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Summer Research Fellowship Programme of India's Science Academies

Infinite series - theory and applications

Hrushikesh Vinod Pawar

Second Year Bachelor of Science, Fergusson College, Pune 411004

Professor R. Thangadurai

Associate Professor, Department of Mathematics, Harish-Chandra Research Institute, Prayagraj (Allahabad) 211002

Abstract

The Series ∑∞n=1(1n)∑∞n=1(1n) and ∑∞n=1log(n+1n)∑∞n=1log(n+1n) both are divergent. But if we subtract second series from first we get ∑∞n=1[1n−log(n+1n)]∑∞n=1[1n−log(n+1n)] which is convergent and converges to a special real number called as "Euler's Constant (γ)(\gamma)". Also a power series, ∑n=0∞(xnn!)\sum\limits_{n=0}^\infty\left(\frac{x^n}{n!}\right) converges to exe^x, for all values of xx.These are some of the series whose convergence or divergence is known to us, but given an arbitrary infinite series, of constant terms or variable terms, how do you find out whether it is convergent or divergent. And if it is convergent then, to which real value does it converge to. These are the two simple looking problems which have driven the whole research of Infinite Series. The first problem among the two is easier to answer because there are different, well defined, methods and techniques, such as a variety of different tests, which can be used to attack these series and establish their convergence or divergence. Of course, the series with positive terms, arbitrary constant terms, alternating series, power series, etc, all have different tests and methods and therefore it's important to study the theory regarding the development of these different tests. The latter problem is not quite easy to solve, as there are not many well-defined methods to find the limiting value. To understand this, consider the Euler's Constant γ\gamma, it is still an open problem about the nature of this real number, we even still don't know whether it is rational number or irrational number, such is the difficulty. These Infinite Series, their convergence or divergence, their limiting values, the rate of their convergence or divergence has a very important place in the mathematics, as they have applications in many different branches of the subject. Also, these series and their limiting value play a very important role and have many applications in the field of Theoretical Physics. This paper is based on developing and understanding this theory from scratch, from the very basics and establishing the proper foundation of the theory and then moving on to tackle some advanced problems and topics like approximating the limiting values of these series, changing the rate of convergence or divergence of a given series, constructing different series converging to same limiting value, etc.

Keywords: tests of convergence, limiting value, power series, rate of convergence, transformation of series, numerical valuations

INTRODUCTION

This paper is based on work done from the book “Theory and Application of Infinite Series”, by Konrad Knopp and some other supplementary books. It is considered that the reader has the basic knowledge of real number system, and some basics terminologies of real analysis. In the book, Dr. Knopp has represented the idea of real numbers as nest of intervals with rational numbers as endpoints, i.e nested intervals of rational numbers. This same idea has been used in some of the proofs. If the reader is not familiar with representing real numbers as nested intervals, it is advisable to read the first chapter from the book by Dr. Knopp.

“Given any sequence, examining their construction shows that there always exists two or more forces, which oppose one another and thereby call forth the variations of terms. One force tends to increase and others to diminish them and it is not clear at a glance which of the two will get the upper hand or in what degree this will happen.”

The paper is divided into 55 sections, first being based on the sequences of Real numbers. It is quite basic and cover the theory of sequences and introduces the reader to idea of infinite sequences and then to Infinite Series, the topic of the paper.
The next two sections focus on infinite series of positive terms and then arbitrary terms and aim to solve Problem AA i.e the convergence or divergence of series. We develop the required theory, study about the algebra of infinite series and define prove lots of tests for the convergence of these series.
Then we have a section on power series, where we define these series, have a look at their properties and the algebra of this series.

In the last section of the paper, we deal with Problem BB i.e developing methods and process that would be helpful to find the limiting value of convergent series and also some part on approximating this values.
The language of the whole paper is easy to understand. All the theorems and proofs in this paper are self contained, and an attempt is made to make use of as many basic arguments as possible in the proofs. I hope the reader enjoys reading this paper and get an overview and a good understanding of the Infinite Series. This paper will surely arm the reader with basic knowledge of Infinite series in general

SEQUENCES OF REAL NUMBERS

Arbitrary Sequences and Arbitrary Null Sequences

We will start by writing down some definitions

Defintion 2.1.1 : If each positive integer 1,2,3,⋯1,2,3,\cdots corresponds to a definite real number xnx_n then the numbers x1,x2,⋯ ,xn,⋯x_1,x_2,\cdots,x_n,\cdots are said to form a sequence.

Definition 2.1.2 : (Bounded Sequence ) - A sequence (xn)(x_n) is said to be bounded if and only if there exists a constant number KK such that ∣xn∣<K,  ∀n\mid x_n\mid < K, \ \ \forall n

Definition 2.1.3 : (Monotonic Sequence) - A sequence (xn)(x_n) is said to be monotonically increasing if xn≤xn+1,  ∀ nx_n \leq x_{n+1}, \ \ \forall \, n and is monotonically decreasing if xn+1≤xn,  ∀ nx_{n+1} \leq x_n, \ \ \forall \, n.

Definition 2.1.3 : (Null Sequence) - A sequence (xn)(x_n) is said to be a null sequence if for every ϵ>0\epsilon > 0, there exists a number n0=n0(ϵ)n_0 = n_0(\epsilon) such that, ∣xn∣<ϵ,  ∀ n>n0\mid x_n\mid < \epsilon, \ \ \forall \, n>n_0.

From the above definition we can say that every null sequence is a bounded sequence (and not vice-versa). In other words we can say that, if for a given sequence (xn)(x_n), the ϵ\epsilon-neighbourhood of 00 contains infinitely many of terms of , then (xn)(x_n) is said to be a null sequence.

Now, as we have defined null sequence we will prove some theorems regarding it.

Theorem 2.1.1: (Comparison test for null sequences) - If (xn)(x_n) is a null sequence and the terms of the sequence (xn′)(x_n'), for every nn beyond a certain mm, satisfies the condition ∣xn′∣<∣xn∣\mid x_n'\mid < \mid x_n\mid or more generally ∣xn′∣<K∣xn∣\mid x_n'\mid < K\mid x_n\mid , where KK is a arbitrary fixed positive number, then (xn′)(x_n') is also a null sequence.

Proof : The proof can be quite easily laid down.
We know that for every ϵ>0  ∃ n0∈N\epsilon > 0 \ \ \exists \,n_0\in \mathbb{N} such that ∣xn∣<ϵ  ∀ n>n0\mid x_n \mid < \epsilon \ \ \forall\, n > n_0.
Now we can choose n0>mn_0 > m, such that for all n>n0,    ∣xn∣<ϵK,n>n_0,\;\;\mid x_n\mid<\frac\epsilon K, where K>0K>0.
But for all these values of n,  ∣xn′∣<∣xn∣<ϵ,  ∀n>n0  ⟹  (xn′)n,\ \ \mid x_n'\mid < \mid x_n\mid < \epsilon, \ \ \forall n > n_0 \implies (x_n') is a null sequence.
Q.E.D​

Theorem 2.1.2: If (xn)(x_n) is a null sequence and (an)(a_n) is any arbitrary bounded sequence then (xn′)=(anxn)(x_n') = (a_nx_n) also forms a null sequence.

Proof: Let ∃ K>0\exists \,K>0 such that ∣an∣<K,  ∀n\mid a_n \mid < K, \ \ \forall n. We can now choose a number n0∈Nn_0 \in \mathbb{N} such that, for every n>n0,  ∣xn∣<ϵ/Kn>n_0, \ \ \mid x_n \mid < \epsilon / K.
Now (xn′)=(anxn)(x_n') = (a_nx_n). Hence for every n>n0,  ∣xn′∣<ϵ  ⟹  (xn′)n>n_0, \ \ \mid x_n' \mid < \epsilon \implies (x_n') is a null sequence.
Q.E.D​

Theorem 2.1.3: If   (xn)(x_n) is a null sequence, then every sub-sequence   (xn′)(x_n') of   (xn)(x_n) is a null sequence.

Proof : Let  (xn′)(x_n') is a subsequence of   (xn)(x_n).
∴  xn′=xkn,  kn\therefore \ \ x_n' = x_{k_n}, \ \ k_n  is some positive integer.

Now, for any   ϵ>0,  ∃ n0∈N, \epsilon > 0,\ \ \exists \, n_0\in \mathbb{N} ,  such that ∀ n>n0,  ∣xn∣<ϵ, \forall\,n>n_0, \ \ |x_n| < \epsilon,  then for any such   n, ∣xn′∣=∣xkn∣<ϵn ,  \mid x_n' \mid = \mid x_{k_n} \mid < \epsilon,   since   kn>n0k_n > n_0,  certainly, if   n>n0n > n_0 .
Q.E.D​

Theorem 2.1.4: Let an arbitrary sequence   (xn)(x_n) be separated into two sub-sequences   (xn′)(x_n') and   (xn′′)(x_n''), so that, every term of   (xn)(x_n) belongs to one and only one of these sub-sequences. If (xn′)(x_n') and   (xn′′)(x_n'') are both null sequences, then so is (xn)(x_n)  itself.

Proof : For a any given  ϵ>0,  ∃ n1, n2∈N \epsilon > 0, \ \ \exists \, n_1, \, n_2 \in\mathbb{N} , such that ∣xn′∣<ϵ,  ∀n>n1\mid x_n' \mid < \epsilon, \ \ \forall n > n_1  and   ∣xn′′∣<ϵ,  ∀ n>n2\mid x_n'' \mid < \epsilon, \ \ \forall\, n > n_2
Now all the elements of   (xn′)(x_n')  and   (xn′′)(x_n'') will be arranged in a definite order in (xn)(x_n)  i.e they would have definite indices and let the terms with indices n1 & n2n_1\, \& \,n_2, now has indices N1 & N2N_1\,\&\,N_2 in (xn)(x_n).
If   n0n_0  is a positive integer   n0>N1  &  n0>N2n_0 > N_1  \,\&\,  n_0 > N_2  then   ∣xn∣<ϵ,  ∀ n>n0\mid x_n \mid < \epsilon, \ \ \forall \, n> n_0.
Q.E.D​​

Theorem 2.1.5: If   (xn)(x_n) is a null sequence and (xn′)(x_n')  is an arbitrary rearrangement of it, then (xn′)(x_n') is also a null sequence.

Proof: For any   ϵ>0  ∃ n0∈N\epsilon > 0 \,\, \exists\, n_0\in\mathbb{N} such that ∀ n>n0,  ∣xn∣<ϵ\forall\,n>n_0,\ \ \mid x_n \mid<\epsilon
From the indices of   x1,x2,⋯ ,xn0x_1,x_2, \cdots, x_{n_0}  obtained after rearrangement, let   n′n'  be
the largest.
Then it's logical to see that for every n>n′,  ∣xn′∣<ϵ  ⟹  (xn′)n>n', \ \ \mid x_n' \mid < \epsilon\implies (x_n')   is also a null sequence.
Q.E.D​​

Theorem 2.1.6: If   (xn)(x_n) is a null sequence and (xn′)(x_n') is obtained by any finite number of
alterations, then (xn′)(x_n') is also a null sequence.

Proof: After doing some finitely many alterations to (xn)(x_n), we have after some   nn  onwards,   xn′=xn+p,    p∈Nx_n'=x_{n+p},\;\;p\in\mathbb{N}
If every (xn)(x_n), for  n≥n1 n \geq n_1  has not changed in (xn′)(x_n') and   n1n_1  has received index   n′n'  then there exists  n0>n′ n_0 > n'  such that   for any given ϵ>0,  ∣xn′∣<ϵ,  ∀ n>n0\epsilon > 0,\ \ \mid x_n' \mid < \epsilon, \, \, \forall\, n > n_0.
Q.E.D​​

Theorem 2.1.7: If   (xn′)(x_n') and  (xn′′) (x_n'') are two null sequences and if the sequence (xn)(x_n) is so related to them by   xn′<xn<xn′′,   ∀ n>m x_n' < x_n < x_n'', \, \,  \forall\, n>m , i.e certain  m -onwards, then   (xn)(x_n) is also a null sequence.

Now, as we understand the null sequences better, let's have a look at the algebra of null sequences.

Theorem 2.1.8: If (xn)(x_n) and (xn′)(x_n') are two null sequences then (yn)=(xn+xn′) (y_n) = (x_n + x_n')  is also a null sequence. (This theorem briefly means, two null sequences can be added together.)

Proof : For any arbitrary ϵ>0,  ∃ n1 & n2\epsilon >0, \ \ \exists\, n_1\, \& \, n_2 such that,​
∣xn∣<ϵ2, ∀n>n1 & ∣xn′∣ϵ2, ∀n>n2\mid x_n \mid < \frac{\epsilon}{2},\, \forall n>n_1 \, \&\, \mid x_n' \mid \frac{\epsilon}{2},\, \forall n > n_2​
Now, let n0n_0 be a number such that n0>n1 & n0>n2n_0 >n_1 \, \& \, n_0>n_2,
hence ∀n>n0 ∣yn∣ = ∣xn+xn′∣<ϵ    ⟹  (yn)\forall n>n_0\, \mid y_n\mid\, = \,\mid x_n+x_n'\mid <\epsilon \ \ \implies (y_n) is a null sequence.
Q.E.D​​

Theorem 2.1.9: If (xn)(x_n) and (xn′)(x_n') are two null sequences then (yn)=(xn−xn′) (y_n) = (x_n - x_n')  is also a null sequence. (Similarly, this theorem briefly means, two null sequences can be subtracted together.)

Proof: We know that if (xn)(x_n) is a null sequence and let cc be any constant real number then, c⋅(xn)=(c⋅xn)c\cdot(x_n) = (c\cdot x_n) is also a null sequence.
Hence, choosing c=−1c=-1, we get −(xn′)=(−xn′)-(x_n') = (-x_n') is also a null sequence.
Then it follows from above theorem, that (yn)=(xn−xn′)(y_n) = (x_n - x_n') is also a null sequence.
Q.E.D

Similarly, the two null sequences can be multiplied term-wise.  But division is not that straight forward. To check, consider (xn/xn)=1(x_n/x_n) = 1 where (xn)\left(x_n\right) is non-zero for all nn, is not a null sequence. Also if (xn)(x_n) is a null sequence, nothing can be said about.

Theorem 2.1.10: If (∣xn∣)(\vert x_n\vert), a sequence of absolute values of terms of (xn)(x_n), has a positive lower bound i.e a number γ>0\gamma > 0 exists, such that, for every n, ∣xn∣ ≥γ>0 n,\, \mid x_n \mid\, \geq \gamma > 0 , then the sequence of reciprocals, (1xn)\left(\frac{1}{x_n}\right), is bounded.

Proof : One can easily see that if ∣xn∣ >γ\mid x_n\mid\,>\gamma then for
k>1γ,  1∣xn∣=∣1xn∣≤k,  ∀ n    ⟹    (1xn)k > \frac{1}{\gamma} ,   \frac{1}{\mid x_n \mid} = \left| \frac{1}{x_n} \right| \leq k, \ \ \forall \, n\ \ \implies \ \ \left(\frac{1}{x_n}\right) is bounded .

The Convergent Sequences

If one reads the previous section carefully, then it can be observed that we have already seen the convergent sequences or rather a special calss of these sequences. Yes, Null Sequences are special class of Convergent Sequences, which converges to 00. Now we will define, what do we mean by convergent sequences in general and then discuss in depth about them.

Definition 2.2.1: If (xn)(x_n) is a given sequence, and if it is related to a definite number ξ\xi in such a way that (xn−ξ)(x_n - \xi) forms a null sequence, then we say that the sequence (xn)(x_n)converges to ξ\xi, or that it is convergent. The number ξ\xi is called the limiting value or limit of this sequence; the sequence is also said to converge to ξ\xi and we say that it's terms approach the (limiting) value ξ\xi , tend to ξ\xi, have the limit ξ\xi. This fact is expressed by the symbols xn→ξx_n \rightarrow \xi as n→∞n\rightarrow\infty or lim⁡n→∞xn=ξ\lim\limits_{n\rightarrow\infty} x_n = \xi.

Or in other words, which convey the same meaning,

Definition 2.2.2 : A sequence X=(xn)X=(x_n) is said to converge to x∈Rx\in\mathbb{R} , or xx is said to be a limit of (xn)(x_n), if for every ϵ>0\epsilon > 0 there exists a natural number KK such that for all n≥Kn\geq K, the terms (xn)(x_n) satisfy, ∣xn−x∣<ϵ\mid x_n-x\mid < \epsilon .

Now as we have defined Convergent Sequences, let us also define the other set of sequences which compliments the set of convergent sequences.

Definition 2.2.3: Any sequence which is not convergent in the sense of definition 2.2.1 or 2.2.2 is called as Divergent Sequence.

Now the class of divergent sequences is further divided into two types - Definitely Divergent and Indefinitely Divergent Sequences.

Definition 2.2.4: If a sequence (xn)(x_n) has a property that for an arbitrary positive number G>0G>0, there exists another number n0n_0 such that for every n>n0, xn>Gn>n_0,\,x_n>G . Then we say that (xn)(x_n) diverges to +∞+\infty, tends to +∞+\infty or is definitely divergent with limit +∞+\infty. We write xn→+∞ (n→∞)x_n \rightarrow +\infty\, (n\rightarrow\infty) or lim⁡xn=+∞\lim x_n = +\infty or lim⁡n→∞xn=+∞ \lim\limits_{n\rightarrow\infty} x_n = +\infty ​

By merely interchanging right and left we get,

Definition 2.2.6: A sequence (xn)(x_n) converges in sense of definition 2.2.1 or 2.2.2 or diverges definitely in sense of definition 2.2.4 or 2.2.5 are said to be definite. All other sequences which neither converge nor diverge definitely are called as indefinitely divergent or indefinite.

Eg: (−1)n,(−2)n,an(a<−1)(-1)^n,(-2)^n,a^n(a<-1)
Now, let xn=(−1)nn,xn→0x_n = \frac{(-1)^n}{n}, x_n\rightarrow 0 as n→∞n\rightarrow \infty, but it converges indefinitely to zero.

One important notation that will be useful further.

Definition 2.2.7: If two sequence (xn)(x_n) and  (yn) (y_n), not necessarily convergent, are related to one another such that the quotient xnyn\frac{x_n}{y_n} tends to a definite finite limiting value (limit) for n→∞n\rightarrow\infty, different from zero, and we write xn∼ynx_n \sim y_n. 

In particular, if this limit is 11 , then we say that the sequences are asymptotically equal write as xn≅ynx_n \cong y_n.

For example : n2+1≅n,  log⁡(5n2+23)∼log⁡n,  12+22+⋯+n2≅13n3\sqrt{n^2+1}\cong n, \ \ \log(5n^2+23)\sim\log n, \ \ 1^2+2^2+\cdots+n^2 \cong \frac{1}{3n^3}

​ Theorems on Convergence of Sequences​‾\underline{Theorems  on  Convergence  of  Sequences​}​

Theorem 2.2.1: A convergent sequence determines its limit quite uniquely.

Proof: If xn→ξx_n \rightarrow \xi and xn→ξ′x_n\rightarrow\xi', simultaneously, then (xn−ξ)(x_n-\xi) and (xn−ξ′)(x_n-\xi') both are null sequences and from theorem 2.1.9 ((xn−ξ)−(xn−ξ′))=(ξ−ξ′)((x_n-\xi)-(x_n-\xi')) = (\xi-\xi') is also a null sequence.

Therefore ξ=ξ′\xi=\xi'
Q.E.D

Theorem 2.2.2: A convergent sequence (xn)(x_n) is invariably, bounded and if ∣xn∣≤K,K>0 |x_n|\leq K, K>0 then for limit ξ,∣ξ∣≤K\xi , |\xi|\leq K.
[Every convergent sequence is a bounded sequence.]

Proof: If xn→ξx_n\rightarrow\xi , then for ϵ>0, ∃ m\epsilon > 0,\, \exists\, m , such that
∀n>m, ∣xn−ξ∣<ϵ\forall n>m, \, |x_n-\xi|<\epsilon ⇒ξ−ϵ<xn<ξ+ϵ\Rightarrow\xi-\epsilon\lt x_n\lt\xi+\epsilon​
Now let KK be a bound on ∣xn∣|x_n| , i.e ∣xn∣<K, ∀ n  \mid x_n\mid\lt K,\,\forall\,n\;
Let K1>xmK_1>x_m, then ∣ξ∣<k1  ⟹  ∣xn∣<K, ∀ n \mid\xi\mid\lt k_1\implies |x_n|\lt K_,\,\forall\,n 
If ∣ξ∣>K  ⟹  ∣ξ−K∣>0|\xi|>K \implies |\xi-K|>0. After some n0n_0 we would have ,
∣xn∣−∣ξ∣≤∣xn−ξ∣<∣ξ∣−K  ⟹  K<∣xn∣|x_n| - |\xi| \leq |x_n-\xi|<|\xi|-K \implies K<|x_n| a contradiction. Hence, ∣ξ∣<K|\xi|\lt K.
Q.E.D

Corollary 2.2.2.1: If xn→ξx_n\rightarrow\xi then ∣xn∣→∣ξ∣\vert x_n\vert\rightarrow|\xi|​

Proof: Now, ∣∣xn∣−∣ξ∣∣≤∣xn−ξ∣||x_n|-|\xi||\leq|x_n-\xi|, hence by comparison test, ∣xn∣→∣ξ∣|x_n|\rightarrow|\xi|.
Q.E.D 

Theorem 2.2.3 : If a convergent sequence (xn)(x_n)has all it's terms different from zero and if it's limit ξ≠0\xi\neq 0 then (1xn)\left(\frac{1}{x_n}\right) is bounded or in other words, a number γ>0\gamma>0,exists such that ∣xn∣≥γ>0|x_n|\geq\gamma>0 for every nn , the numbers ∣xn∣|x_n| possesses a positive lower bound.​

Theorem 2.2.4: If (xn′)(x_n') is a subsequence of (xn)(x_n)then xn→ξ  ⟹  xn′→ξx_n\rightarrow\xi\implies x_n'\rightarrow\xi​

Theorem 2.2.5: If a sequence (xn)(x_n)can be divided into two sub-sequences of which each converges to ξ\xi then (xn)(x_n) also converges to ξ\xi.

Theorem 2.2.6: If (xn′)(x_n')is an arbitrary rearrangement of (xn)(x_n)then (xn)→ξ⇒(xn′)→ξ(x_n)\rightarrow\xi\Rightarrow(x_n')\rightarrow\xi​

Theorem 2.2.7: If (xn)→ξ(x_n)\rightarrow\xi and if (xn′)(x_n') is obtained by finite alterations of (xn)(x_n), then (xn′)→ξ(x_n')\rightarrow\xi​

Theorem 2.2.8: If (xn′)→ξ(x_n')\rightarrow\xi and (xn′′)→ξ(x_n'')\rightarrow\xiand if sequence (xn)(x_n)lies between (xn′)(x_n') and (xn′′)(x_n'')from some mm onwards i.e ∀ n>m\forall\,n>m, then xn→ξx_n\rightarrow\xi​

The proofs of the above theorems follows directly from the proof the similar theorems for null sequences in the last section. Next we will see some theorems regarding the calculations concerning convergent series.

​ Calculations with Convergent Sequences​‾\underline{Calculations  with  Convergent  Sequences​}​

Theorem 2.2.9: If xn→ξx_n\rightarrow\xi and yn→ηy_n\rightarrow\eta always implies (xn+yn)→ξ+η(x_n+y_n)\rightarrow\xi+\eta and the statement holds for term by term addition for any fixed number - say P - of convergent sequences.

Proof: As (xn−ξ)(x_n-\xi) and (yn−η)(y_n-\eta) are null sequences, from theorem 2.1.8, we know that ((xn+yn)−(ξ+η))((x_n+y_n)-(\xi+\eta)) is also a null sequences.
Q.E.D 

Corollary 2.2.9.1: xn→ξx_n\rightarrow\xi and yn→ηy_n\rightarrow\eta always implies (xn−yn)→ξ−η(x_n-y_n)\rightarrow\xi-\eta​

Proof follows from theorem 2.1.9
Q.E.D 

Theorem 2.2.10: xn→ξx_n\rightarrow\xi and yn→ηy_n\rightarrow\eta always implies (xn⋅yn)→ξ⋅η(x_n\cdot y_n)\rightarrow\xi\cdot\eta and the statement holds for term by term multiplication for any fixed number - say P - of convergent sequences.

Proof: We have xn⋅yn−ξ⋅η=yn⋅(xn−ξ)−(yn−η)⋅ξx_n\cdot y_n-\xi\cdot\eta = y_n\cdot(x_n-\xi)-(y_n-\eta)\cdot\xi. And now it follows from theorem 2.1.2, that (xn⋅yn−ξ⋅η)(x_n\cdot y_n-\xi\cdot\eta) forms a null sequence.
Q.E.D 

Theorem 2.2.11: xn→ξx_n\rightarrow\xi and yn→ηy_n\rightarrow\eta always implies, if every yn≠0y_n=\not0 and also η≠0\eta=\not0 then (xnyn)→ξη\left(\frac{x_n}{y_n}\right)\rightarrow\frac{\xi}{\eta}​

Proof: We have xnyn−ξη=xn⋅η−yn⋅ξyn⋅η=(xn−ξ)⋅η−(yn−η)⋅ξyn⋅η\frac{x_n}{y_n}-\frac{\xi}{\eta} = \frac{x_n\cdot\eta-y_n\cdot\xi}{y_n\cdot\eta}=\frac{(x_n-\xi)\cdot\eta-(y_n-\eta)\cdot\xi}{y_n\cdot\eta} 
Now from above theorem 2.2.10, the numerator is a null sequence and from theorem 2.2.3, (xnyn)\left(\frac{x_n}{y_n}\right) is a bounded sequence. Hence finally using theorem 2.1.2, our theorem is proved.
Q.E.D 

 

​ Cauchy’s Theorems of limit and its Generalization‾\underline{Cauchy’s  Theorems  of  limit  and  its  Generalization}

All the theorems which we have done till now, contibute to the basic theory of Convergent Sequences. Now we will see some important ones, given by Cauchy, which will be helpful to us in our long run.

Theorem 2.2.12: If xn=(x0,x1,⋯ ,xn)x_n=(x_0,x_1,\cdots,x_n) is a null sequence, then the arithmetic mean (xn′)=x0+x1+⋯+xnn+1(x_n') = \frac{x_0+x_1+\cdots+x_n}{n+1} is also a null sequence.

Proof: For any given ϵ>0, ∃ m\epsilon>0,\,\exists\,m such that ∣xn∣<ϵ/2, ∀ n>m|x_n|<\epsilon/2,\,\forall\,n>m.
For these nn's, we have ∣xn′∣<x0+⋯+xmn+1+n−mn+1⋅ϵ2|x_n'|<\frac{x_0+\cdots+x_m}{n+1} + \frac{n-m}{n+1}\cdot\frac{\epsilon}{2} (Every term >xm>x_m is <ϵ2<\frac{\epsilon}{2})
The first fraction above is formed of finitely many terms, hence we can choose an n0>mn_0>m such that for every n>n0, x0+⋯+xmn+1<ϵ2  ⟹  ∣xn′∣<ϵ, ∀ n>n0  ⟹  (xn′)n>n_0,\,\frac{x_0+\cdots+x_m}{n+1}<\frac{\epsilon}{2}\implies |x_n'|<\epsilon,\,\forall\,n>n_0\implies(x_n')is a null sequence.
Q.E.D 

Corollary 2.2.12.1: If xn→ξx_n\rightarrow\xi then so does the arithmetic mean, (xn′)=x0+x1+⋯+xnn+1(x_n') = \frac{x_0+x_1+\cdots+x_n}{n+1}​

Proof: We have ((x0−ξ)+(x1−ξ)+⋯+(xn−ξ)n+1)=(x0+⋯+xnn+1−(n+1)ξn+1)=(xn′−ξ)\left(\frac{(x_0-\xi)+(x_1-\xi)+\cdots+(x_n-\xi)}{n+1}\right) = \left(\frac{x_0+\cdots+x_n}{n+1}-\frac{(n+1)\xi}{n+1}\right)=(x_n'-\xi).
From the above theorem (xn′−ξ)(x_n'-\xi) is a null sequence when (xn−ξ)(x_n-\xi) is i.e xn→ξ  ⟹  xn′→ξ x_n\rightarrow\xi\implies x_n'\rightarrow\xi  
Q.E.D

Now a similar theorem for the Geometric Mean.

Theorem 2.2.13: Let yn=(y1,⋯ ,yn)→ηy_n=(y_1,\cdots,y_n)\rightarrow\eta and the terms in it and η\eta be positive. Then the Geometric Mean (yn′)=y1⋅y2⋯ynn→η(y_n')=\sqrt[n]{y_1\cdot y_2\cdots y_n}\rightarrow\eta i.e (yn′−η)(y_n'-\eta)​is a null sequence.

Proof: Let (xn)=log⁡yn′(x_n)=\log y_n'
Now using properties of log⁡\log we get xn=log⁡yn→ξ=log⁡η.x_n = \log y_n\rightarrow\xi=\log\eta .
∴ (xn′)=x1+x2+⋯+xnn=log⁡y1⋯ynn\therefore  (x_n') =\frac{x_1+x_2+\cdots+x_n}{n} = \log\sqrt[n]{y_1\cdots y_n}
By using the above corollary 2.2.11.1, we get, log⁡y1⋯ynn→log⁡η  ⟹  y1⋯ynn→η\log\sqrt[n]{y_1\cdots y_n}\rightarrow\log\eta \implies\sqrt[n]{y_1\cdots y_n}\rightarrow\eta​
Q.E.D 

Now we weil move a step forward and try to generalize the above theorems so that we can use them afterwards in more general setting.

Generalization of Cauchy’s Theorems‾\underline{\textit{Generalization of Cauchy’s Theorems}}

Theorem 2.2.14: Let (x0,⋯ ,xn)(x_0,\cdots,x_n) be a null sequence and let the coefficients of the following system, satisfy the following two conditions :

A{a00a10  a11a20  a21  a22⋮an0  an1  ⋯  ann\displaystyle \begin{aligned}A \begin{cases}&a_{00}\\&a_{10}\ \ a_{11}\\&a_{20}\ \ a_{21}\ \ a_{22}\\&\vdots\\&a_{n0}\ \ a_{n1}\ \ \cdots \ \ a_{nn}\end{cases}\end{aligned}
  • Every column contains a null sequence i.e for a fixed p≥0, anp→0p\geq0,\,a_{np}\rightarrow0 as n→∞ n\rightarrow\infty 
  • There exits a constant KK , such that, sum of the absolute values of terms in any one row i.e for every n n ,  ∣an0∣+∣an1∣+⋯+∣ann∣  |a_{n0}|+|a_{n1}|+\cdots+|a_{nn}| 

Then the sequence formed by numbers xn′=an0x0+an1x1+⋯+annxn x_n' = a_{n0}x_0 + a_{n1}x_1+\cdots+a_{nn}x_n  is also a null sequence.

Proof: For any given ϵ>0, ∃ m\epsilon>0,\,\exists\,m such that ∣xn∣<ϵ, ∀ n>m|x_n|<\epsilon,\,\forall\,n>m.
∴ ∣xn′∣<∣an0x0+an1x1+⋯+anmxm∣+ϵ2\therefore  |x_n'|<|a_{n0}x_0 + a_{n1}x_1+\cdots+a_{nm}x_m| + \frac{\epsilon}{2}
From condition (1) we may now choose n0>mn_0>m such that,
∣an0x0+an1x1+⋯+anmxm∣<ϵ2, ∀ n>n0  ⟹  ∣xn∣<ϵ,∀ n>n0|a_{n0}x_0 + a_{n1}x_1+\cdots+a_{nm}x_m|<\frac{\epsilon}{2},\,\forall\,n>n_0 \implies |x_n|<\epsilon,\forall\,n>n_0 ​
Q.E.D 

Corollary 2.2.14.1: If our coefficients aλμa_{\lambda\mu} are substituted by any other numbers aλμ′=αμaλμa_{\lambda\mu}'=\alpha_{\mu}a_{\lambda\mu}, obtained from from numbers aλμ a_{\lambda\mu}  by multiplication by factor  αλμ  \alpha_{\lambda\mu} , all in absolute values less than fixed  α  \alpha , then the numbers  xn′′=an0′x0+⋯+ann′xn  x_n''= a_{n0}'x_0+\cdots+a_{nn}'x_n  also forms a null sequence.

Proof: The aλμ′a'_{\lambda\mu} satisfies conditions (1) and (2) in above theorem, hence for a fixed p≥0p\geq 0, anp′→0a_{np}' \rightarrow 0, by theorem 2.1.1. Also ∣an0′∣+∣an1′∣+⋯+∣ann′∣<αK|a_{n0}'|+|a_{n1}'|+\cdots+|a_{nn}'|<\alpha K. Hence applying theorem 2.1.2, we get the desired result. ​
Q.E.D 

Theorem 2.2.15: If (xn)→ξ(x_n)\rightarrow\xi and coefficient aμηa_{\mu\eta} beside the condition (1) and (2), satisfy, a new condition
c. An=an0+⋯+ann→1 A_n = a_{n0}+\cdots+a_{nn} \rightarrow 1 ,
then all also the numbers  xn′=an0xn+⋯+annxn→ξ  x_n' = a_{n0}x_n+\cdots+a_{nn}x_n\rightarrow\xi 

Proof: Let xn′=Anξ+ann0(x0−ξ)+⋯+ann(xn−ξ)x_n' = A_n\xi + a_{nn_0}(x_0-\xi) +\cdots+a_{nn}(x_n-\xi)
(xn−ξ)(x_n-\xi)is a null sequence and (ann)(a_{nn}) is a bounded sequence and (An−1)(A_n-1) is a null sequence, while ξ\xi is a constant. Then using theorems 2.1.1 and 2.1.2 (xn′)(x_n') is also a null sequence. ​
Q.E.D 

Two Main Criteria

So far, we have developed enough theory to ask and ponder on questions realted to sequences. Now, we face to great problems :
(A):(A) : Is a given sequence (xn)(x_n) convergent or divergent (definitely or indefinitely).
This means, if presented with any sequence, how can one deduce whether it is convergent or divergent. If we somehow prove that given sequence is convergent, then comes the question of its limiting value. (B):(B) : To what limit ξ\xi, does the sequence (xn)(x_n)recognized to be convergent, tend?

At this moment the task of finding the limiting value might almost seem insolvable or else trivial. Now trivial because, according to our uniqueness theorem, each convergent sequence entirely determines its limit ξ\xi i.e it is the number itself and the solution may be considered as given, a real number represented in the form of a sequence.

Hence, an another way of stating our problem (B) will be

​ (B′):(B'): Two convergent sequences (xn)(x_n)and (xn′)(x_n') are given, how may we determine whether or not both define same limit or whether or not the two limits stand in simple relation to one another.

Now the focus of the remaining paper will be to answer this problems. But for the time being, we will leave problem B or B' untouched and focus all our attention and power in developing weapons(tools) and defenses(theory) to defeat problem A.

First Main Criterion (For Monotone Sequences)‾\underline{\textit{First Main Criterion (For Monotone Sequences)}}

Theorem 2.3.1: A monotone bounded sequence is invariably convergent and a monotone sequence which is not bounded is always definitely divergent.

Proof: Here we have to prove two things :
a) If (xn)(x_n) is increasing and unbounded then (xn)→∞(x_n)\rightarrow\infty and similarly if (yn)(y_n)is decreasing and unbounded, (yn)→−∞(y_n)\rightarrow-\infty
b) If (xn)(x_n) is increasing and bounded then xn→ξx_n\rightarrow\xi similar case for decreasing sequences.

Part A: If a sequence (xn)(x_n) is monotonically increasing then xn+1>xn, ∀ nx_{n+1}>x_n,\,\forall\,n. Hence the sequence is bounded below (or on the left) as xn≥x1, ∀ nx_{n}\geq x_1,\,\forall\,n.
Consider (xn)(x_n)is not bounded on right, i.e for any given G>0G>0, we always find an index n0n_0 for which xn0>G  ⟹  xn>G, ∀ n>n0x_{n_0}>G\implies x_n>G,\,\forall\,n>n_0
Hence by definition 2.2.4, xn→+∞x_n \rightarrow +\infty
Similarly by some interchanges we can prove that a monotonically decreasing sequence (yn)(y_n)not bounded, yn→−∞ y_n\rightarrow-\infty ​

Part B: Now let (xn)(x_n)be a monotonically increasing bounded sequence. i.e ∃ K\exists\, K such that x1≤xn<K, ∀ nx_1\leq x_n \lt K,  \forall  n
The interval J1=(x1,K)J_1=(x_1,K) contains all the terms of (xn)(x_n). We will apply method of successive bisection to this interval and after bisecting the interval we denote right or left half of J1J_1as J2J_2depending on whether it does or doesn't contains points of (xn)(x_n)

We apply the same rules on J2J_2 and construct another interval J3J_3 and so on. Now for any interval JkJ_kwe can choose nkn_k such that ∀ n>nk\forall\,n>n_k, all such (xn)(x_n)lies to right of JkJ_k and no such term of (xn)(x_n) is to the left. Let ξ\xi be the number determined by the nest (Jn)(J_n).

Now, for any ϵ>0\epsilon>0, we can choose p∈Np\in\mathbb{N}such that the length of JpJ_p is less than ϵ\epsilon .
∴ ∀ n>np\therefore  \forall \,n>n_p, the (xn)(x_n)'s along with ξ\xi resides
Jp  ⟹  ∣xn−ξ∣<ϵ, ∀ n>np  ⟹  (xn−ξ)J_p \implies |x_n-\xi|<\epsilon,\,\forall\,n>n_p\implies (x_n-\xi) in is a null sequence i.e xn→ξx_n\rightarrow\xi.
By making, suitable changes we can prove the same for a bounded monotonically decreasing sequence. ​
Q.E.D 

If we carefully look at the above criteria, though useful, it applies to a restricted class of sequences i.e monotone sequences. We need something which will be applicable on wider scale, nearly to every sequence possible. Hence we have,

Second Main Criterion‾\underline{\textit{Second Main Criterion}}

​Theorem 2.3.2: An arbitrary sequence (xn)(x_n)is said to be Cauchy Sequence if and only if, given  ϵ>0  \epsilon>0 , a natural number  n0=n0(ϵ)  n_0=n_0(\epsilon)  can always be assigned such that for any two indices  n & m   n\,\&\,m  both greater than n0  n_0  , we have  ∣xn−xm∣<ϵ  |x_n-x_m|<\epsilon .
Every Cauchy sequence is convergent and every convergent sequence is a Cauchy Sequence.

Proof:
Part A: It deals with proving that if a sequence is convergent then it follows the above condition.
[In simple words first will prove that every convergent sequence is Cauchy Sequence.]
If (xn)(x_n) is convergent and tends to ξ\xi then for any ϵ>0, ∃ n0\epsilon>0,\,\exists\,n_0 such that ∣xn−ξ∣<ϵ2, ∀ n>n0|x_n-\xi|<\frac{\epsilon}{2},\,\forall\,n>n_0.
Now we choose, n′>n0n'>n_0such that, ∣xn′−ξ∣<ϵ2|x_{n'}-\xi|<\frac{\epsilon}{2}.
∴ ∣xn−xn′∣=∣(xn−ξ)−(xn′−ξ)∣≤∣xn−ξ∣+∣xn′−ξ∣<ϵ\therefore  |x_n-x_{n'}|=|(x_n-\xi)-(x_{n'}-\xi)|\leq|x_n-\xi|+|x_{n'}-\xi|<\epsilon
This proves our first part.

Part B : It concerns with proving that (in simple terms) every Cauchy sequence is convergent. Here we will prove it using nest of intervals.
Consider that given ϵ>0,\epsilon>0 , we can always assign an index pp such that for all indices n>p,∣xn−xp∣<ϵn>p , |x_n-x_p|<\epsilon.

Now to construct a nest of intervals let ϵ=12,14,⋯ ,12k\epsilon = \frac{1}{2},\frac{1}{4},\cdots,\frac{1}{2^k}, we get
1) There is an index p1p_1 such that for n>p1,∣xn−xp1∣<12n>p_1 , |x_n-x_{p_1}|<\frac{1}{2}
2) There exists an index p2p_2 such that for n>p2, ∣xn−xp2∣<14n>p_2,\,|x_n-x_{p_2}|<\frac{1}{4}
3) Going on like this, there exists pkp_k such that ∀n>pk,∣xn−xpk∣<12k\forall n>p_k , |x_n-x_{p_k}|<\frac{1}{2^k}

Accordingly we form intervals JkJ_ksuch that intervals (xp1−12,xp1+12)(x_{p_1}-\frac{1}{2},x_{p_1}+\frac{1}{2}) is called J1J_1. It contains all the (xn)(x_n)'s for which n>p1n>p_1. Hence it contains xp2x_{p_2}

Therefore it also contains whole of interval (xp2−14,xp2−14)(x_{p_2}-\frac{1}{4},x_{p_2}-\frac{1}{4})which contains all (xn)(x_n)'s such that n>p2n>p_2. J2J_2 is an interval formed by common points of J1J_1and this interval.
Going on like this, we construct Jk