Infinite series - theory and applications
Abstract
The Series $n=1∑∞ (n1 )$ and $n=1∑∞ g(nn+1 )$ both are divergent. But if we subtract second series from first we get $n=1∑∞ [n1 −g(nn+1 )]$ which is convergent and converges to a special real number called as "Euler's Constant $(\gamma)$". Also a power series, $\sum\limits_{n=0}^\infty\left(\frac{x^n}{n!}\right)$ converges to $e^x$, for all values of $x$.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 $\gamma$, 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 $5$ 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 $A$ 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 $B$ 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 $1,2,3,\cdots$ corresponds to a definite real number $x_n$ then the numbers $x_1,x_2,\cdots,x_n,\cdots$ are said to form a sequence.
Definition 2.1.2 : (Bounded Sequence ) - A sequence $(x_n)$ is said to be bounded if and only if there exists a constant number $K$ such that $\mid x_n\mid < K, \ \ \forall n$
Definition 2.1.3 : (Monotonic Sequence) - A sequence $(x_n)$ is said to be monotonically increasing if $x_n \leq x_{n+1}, \ \ \forall \, n$ and is monotonically decreasing if $x_{n+1} \leq x_n, \ \ \forall \, n$.
Definition 2.1.3 : (Null Sequence) - A sequence $(x_n)$ is said to be a null sequence if for every $\epsilon > 0$, there exists a number $n_0 = n_0(\epsilon)$ such that, $\mid x_n\mid < \epsilon, \ \ \forall \, n>n_0$.
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 $(x_n)$, the $\epsilon$-neighbourhood of $0$ contains infinitely many of terms of , then $(x_n)$ 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 $(x_n)$ is a null sequence and the terms of the sequence $(x_n')$, for every $n$ beyond a certain $m$, satisfies the condition $\mid x_n'\mid < \mid x_n\mid$ or more generally $\mid x_n'\mid < K\mid x_n\mid$ , where $K$ is a arbitrary fixed positive number, then $(x_n')$ is also a null sequence.
Proof : The proof can be quite easily laid down.
We know that for every $\epsilon > 0 \ \ \exists \,n_0\in \mathbb{N}$ such that $\mid x_n \mid < \epsilon \ \ \forall\, n > n_0$.
Now we can choose $n_0 > m$, such that for all $n>n_0,\;\;\mid x_n\mid<\frac\epsilon K,$ where $K>0$.
But for all these values of $n,\ \ \mid x_n'\mid < \mid x_n\mid < \epsilon, \ \ \forall n > n_0 \implies (x_n')$ is a null sequence.
Q.E.D
Theorem 2.1.2: If $(x_n)$ is a null sequence and $(a_n)$ is any arbitrary bounded sequence then $(x_n') = (a_nx_n)$ also forms a null sequence.
Proof: Let $\exists \,K>0$ such that $\mid a_n \mid < K, \ \ \forall n$. We can now choose a number $n_0 \in \mathbb{N}$ such that, for every $n>n_0, \ \ \mid x_n \mid < \epsilon / K$.
Now $(x_n') = (a_nx_n)$. Hence for every $n>n_0, \ \ \mid x_n' \mid < \epsilon \implies (x_n')$ is a null sequence.
Q.E.D
Theorem 2.1.3: If $(x_n)$ is a null sequence, then every sub-sequence $(x_n')$ of $(x_n)$ is a null sequence.
Proof : Let $(x_n')$ is a subsequence of $(x_n)$.
$\therefore \ \ x_n' = x_{k_n}, \ \ k_n$ is some positive integer.
Q.E.D
Theorem 2.1.4: Let an arbitrary sequence $(x_n)$ be separated into two
Proof : For a any given $\epsilon > 0, \ \ \exists \, n_1, \, n_2 \in\mathbb{N}$ , such that
Now all the elements of $(x_n')$ and $(x_n'')$ will be arranged in a definite order in $(x_n)$ i.e they would have definite indices
If $n_0$ is a positive integer $n_0 > N_1 \,\&\, n_0 > N_2$ then $\mid x_n \mid < \epsilon, \ \ \forall \, n> n_0$.
Q.E.D
Theorem 2.1.5: If $(x_n)$ is a null sequence and $(x_n')$ is an arbitrary rearrangement of it, then $(x_n')$ is also a null sequence.
Proof: For any $\epsilon > 0 \,\, \exists\, n_0\in\mathbb{N}$ such that $\forall\,n>n_0,\ \ \mid x_n \mid<\epsilon$
From the indices of $x_1,x_2, \cdots, x_{n_0}$ obtained after rearrangement, let $n'$ be
the largest.
Then it's logical to see that for every $n>n', \ \ \mid x_n' \mid < \epsilon\implies (x_n')$ is also a null sequence.
Q.E.D
Theorem 2.1.6: If $(x_n)$ is a null sequence and $(x_n')$ is obtained by any finite number of
alterations, then $(x_n')$ is also a null sequence.
Proof: After doing some finitely many alterations to $(x_n)$, we have after some $n$ onwards, $x_n'=x_{n+p},\;\;p\in\mathbb{N}$
If every $(x_n)$, for $n \geq n_1$ has not changed in $(x_n')$ and $n_1$ has received index $n'$ then there exists $n_0 > n'$ such that for any given $\epsilon > 0,\ \ \mid x_n' \mid < \epsilon, \, \, \forall\, n > n_0$.
Q.E.D
Theorem 2.1.7: If $(x_n')$ and $(x_n'')$ are two null sequences and if the
Now, as we understand the null sequences better, let's have a look at the algebra of null sequences.
Theorem 2.1.8: If $(x_n)$ and $(x_n')$ are two null sequences then $(y_n) = (x_n + x_n')$ is also a null sequence. (This theorem briefly means, two null sequences can be added together.)
Proof : For any arbitrary $\epsilon >0, \ \ \exists\, n_1\, \& \, n_2$ such that,
$\mid x_n \mid < \frac{\epsilon}{2},\, \forall n>n_1 \, \&\, \mid x_n' \mid \frac{\epsilon}{2},\, \forall n > n_2$
Now, let $n_0$ be a number such that $n_0 >n_1 \, \& \, n_0>n_2$,
hence $\forall n>n_0\, \mid y_n\mid\, = \,\mid x_n+x_n'\mid <\epsilon \ \ \implies (y_n)$ is a null sequence.
Q.E.D
Theorem 2.1.9: If $(x_n)$ and $(x_n')$ are two null sequences then $(y_n) = (x_n - x_n')$ is also a null sequence. (Similarly, this theorem briefly means, two null sequences can be subtracted together.)
Proof: We know that if $(x_n)$ is a null sequence and let $c$ be any constant real number then, $c\cdot(x_n) = (c\cdot x_n)$ is also a null sequence.
Hence, choosing $c=-1$, we get $-(x_n') = (-x_n')$ is also a null sequence.
Then it follows from above theorem, that $(y_n) = (x_n - x_n')$ 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 $(x_n/x_n) = 1$ where $\left(x_n\right)$ is non-zero for all $n$, is not a null sequence. Also if $(x_n)$ is a null sequence, nothing can be said about.
Theorem 2.1.10: If $(\vert x_n\vert)$, a sequence of absolute values of terms of $(x_n)$, has a positive lower bound i.e a number $\gamma > 0$ exists, such that, for every $n,\, \mid x_n \mid\, \geq \gamma > 0$, then the sequence of reciprocals, $\left(\frac{1}{x_n}\right)$, is bounded.
Proof : One can easily see that if $\mid x_n\mid\,>\gamma$ then for
$k > \frac{1}{\gamma} , \frac{1}{\mid x_n \mid} = \left| \frac{1}{x_n} \right| \leq k, \ \ \forall \, n\ \ \implies \ \ \left(\frac{1}{x_n}\right)$ 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 $0$. 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 $(x_n)$ is a given sequence, and if it is related to a definite number $\xi$ in such a way that $(x_n - \xi)$ forms a null sequence, then we say that the sequence $(x_n)$converges to $\xi$, or that it is convergent. The number $\xi$ is called the limiting value or limit of this sequence; the sequence is also said to converge to $\xi$ and we say that it's terms approach the (limiting) value $\xi$ , tend to $\xi$, have the limit $\xi$. This fact is expressed by the symbols $x_n \rightarrow \xi$ as $n\rightarrow\infty$ or $\lim\limits_{n\rightarrow\infty} x_n = \xi$.
Or in other words, which convey the same meaning,
Definition 2.2.2 : A sequence $X=(x_n)$ is said to converge to $x\in\mathbb{R}$ , or $x$ is said to be a limit of $(x_n)$, if for every $\epsilon > 0$ there exists a natural number $K$ such that for all $n\geq K$, the terms $(x_n)$ satisfy, $\mid x_n-x\mid < \epsilon$ .
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 $(x_n)$ has a property that for an arbitrary positive number $G>0$, there exists another number $n_0$ such that for every $n>n_0,\,x_n>G$ . Then we say that $(x_n)$ diverges to $+\infty$, tends to $+\infty$ or is definitely divergent with limit $+\infty$. We write $x_n \rightarrow +\infty\, (n\rightarrow\infty)$ or $\lim x_n = +\infty$ or $\lim\limits_{n\rightarrow\infty} x_n = +\infty$
By merely interchanging right and left we get,
Definition 2.2.6: A sequence $(x_n)$ 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: $(-1)^n,(-2)^n,a^n(a<-1)$
Now, let $x_n = \frac{(-1)^n}{n}, x_n\rightarrow 0$ as $n\rightarrow \infty$, but it converges indefinitely to zero.
One important notation that will be useful further.
Definition 2.2.7: If two sequence $(x_n)$ and $(y_n)$, not necessarily convergent, are related to one another such that the quotient $\frac{x_n}{y_n}$ tends to a definite finite limiting value (limit) for $n\rightarrow\infty$, different from zero, and we write $x_n \sim y_n$.
In particular, if this limit is $1$ , then we say that the sequences are asymptotically equal write as $x_n \cong y_n$.
For example : $\sqrt{n^2+1}\cong n, \ \ \log(5n^2+23)\sim\log n, \ \ 1^2+2^2+\cdots+n^2 \cong \frac{1}{3n^3}$
$\underline{Theorems on Convergence of Sequences}$
Theorem 2.2.1: A convergent sequence determines its limit quite uniquely.
Proof: If $x_n \rightarrow \xi$ and $x_n\rightarrow\xi'$, simultaneously, then $(x_n-\xi)$ and $(x_n-\xi')$ both are null sequences and from theorem 2.1.9 $((x_n-\xi)-(x_n-\xi')) = (\xi-\xi')$ is also a null sequence.
Therefore $\xi=\xi'$
Q.E.D
Theorem 2.2.2: A convergent sequence $(x_n)$ is invariably, bounded and if $|x_n|\leq K, K>0$then for limit $\xi , |\xi|\leq K$.
[Every convergent sequence is a bounded sequence.]
Corollary 2.2.2.1: If $x_n\rightarrow\xi$ then $\vert x_n\vert\rightarrow|\xi|$
Proof: Now, $||x_n|-|\xi||\leq|x_n-\xi|$, hence by comparison test, $|x_n|\rightarrow|\xi|$.
Q.E.D
Theorem 2.2.3 : If a convergent sequence $(x_n)$has all it's terms different from zero and if it's limit $\xi\neq 0$ then $\left(\frac{1}{x_n}\right)$ is bounded or in other words, a number $\gamma>0$,exists such that $|x_n|\geq\gamma>0$ for every $n$ , the numbers $|x_n|$ possesses a positive lower bound.
Theorem 2.2.4: If $(x_n')$ is a subsequence of $(x_n)$then $x_n\rightarrow\xi\implies x_n'\rightarrow\xi$
Theorem 2.2.5: If a sequence $(x_n)$can be divided into two sub-sequences of which each converges to $\xi$ then $(x_n)$ also converges to $\xi$.
Theorem 2.2.6: If $(x_n')$is an arbitrary rearrangement of $(x_n)$then $(x_n)\rightarrow\xi\Rightarrow(x_n')\rightarrow\xi$
Theorem 2.2.7: If $(x_n)\rightarrow\xi$ and if $(x_n')$ is obtained by finite alterations of $(x_n)$, then $(x_n')\rightarrow\xi$
Theorem 2.2.8: If $(x_n')\rightarrow\xi$ and $(x_n'')\rightarrow\xi$and if sequence $(x_n)$lies between $(x_n')$ and $(x_n'')$from some $m$ onwards i.e $\forall\,n>m$, then $x_n\rightarrow\xi$
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.
$\underline{Calculations with Convergent Sequences}$
Theorem 2.2.9: If $x_n\rightarrow\xi$ and $y_n\rightarrow\eta$ always implies $(x_n+y_n)\rightarrow\xi+\eta$ and the statement holds for term by term addition for any fixed number - say P - of convergent sequences.
Proof: As $(x_n-\xi)$ and $(y_n-\eta)$ are null sequences, from theorem 2.1.8, we know that $((x_n+y_n)-(\xi+\eta))$ is also a null sequences.
Q.E.D
Corollary 2.2.9.1: $x_n\rightarrow\xi$ and $y_n\rightarrow\eta$ always implies $(x_n-y_n)\rightarrow\xi-\eta$
Proof follows from theorem 2.1.9
Q.E.D
Theorem 2.2.10: $x_n\rightarrow\xi$ and $y_n\rightarrow\eta$ always implies $(x_n\cdot y_n)\rightarrow\xi\cdot\eta$ and the statement holds for term by term multiplication for any fixed number - say P - of convergent sequences.
Proof: We have $x_n\cdot y_n-\xi\cdot\eta = y_n\cdot(x_n-\xi)-(y_n-\eta)\cdot\xi$. And now it follows from theorem 2.1.2, that $(x_n\cdot y_n-\xi\cdot\eta)$ forms a null sequence.
Q.E.D
Theorem 2.2.11: $x_n\rightarrow\xi$ and $y_n\rightarrow\eta$ always implies, if every $y_n=\not0$ and also $\eta=\not0$ then $\left(\frac{x_n}{y_n}\right)\rightarrow\frac{\xi}{\eta}$
Proof: We have $\frac{x_n}{y_n}-\frac{\xi}{\eta} = \frac{x_n\cdot\eta-y_n\cdot\xi}{y_n\cdot\eta}=\frac{(x_n-\xi)\cdot\eta-(y_n-\eta)\cdot\xi}{y_n\cdot\eta}$
Now from above theorem 2.2.10, the numerator is a null sequence and from theorem 2.2.3, $\left(\frac{x_n}{y_n}\right)$ is a bounded sequence. Hence finally using theorem 2.1.2, our theorem is proved.
Q.E.D
$\underline{Cauchy’s Theorems of limit and its Generalization}$
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 $x_n=(x_0,x_1,\cdots,x_n)$ is a null sequence, then the arithmetic mean $(x_n') = \frac{x_0+x_1+\cdots+x_n}{n+1}$ is also a null sequence.
Proof: For any given $\epsilon>0,\,\exists\,m$ such that $|x_n|<\epsilon/2,\,\forall\,n>m$.
For these $n$'s, we have $|x_n'|<\frac{x_0+\cdots+x_m}{n+1} + \frac{n-m}{n+1}\cdot\frac{\epsilon}{2}$ (Every term $>x_m$ is $<\frac{\epsilon}{2}$)
The first fraction above is formed of finitely many terms, hence we can choose an $n_0>m$ such that for every $n>n_0,\,\frac{x_0+\cdots+x_m}{n+1}<\frac{\epsilon}{2}\implies |x_n'|<\epsilon,\,\forall\,n>n_0\implies(x_n')$is a null sequence.
Q.E.D
Corollary 2.2.12.1: If $x_n\rightarrow\xi$ then so does the arithmetic mean, $(x_n') = \frac{x_0+x_1+\cdots+x_n}{n+1}$
Proof: We have $\left(\frac{(x_0-\xi)+(x_1-\xi)+\cdots+(x_n-\xi)}{n+1}\right) = \left(\frac{x_0+\cdots+x_n}{n+1}-\frac{(n+1)\xi}{n+1}\right)=(x_n'-\xi)$.
From the above theorem $(x_n'-\xi)$ is a null sequence when $(x_n-\xi)$ is i.e $x_n\rightarrow\xi\implies x_n'\rightarrow\xi$
Q.E.D
Now a similar theorem for the Geometric Mean.
Theorem 2.2.13: Let $y_n=(y_1,\cdots,y_n)\rightarrow\eta$ and the terms in it and $\eta$ be positive. Then the Geometric Mean $(y_n')=\sqrt[n]{y_1\cdot y_2\cdots y_n}\rightarrow\eta$ i.e $(y_n'-\eta)$is a null sequence.
Proof: Let $(x_n)=\log y_n'$
Now using properties of $\log$ we get $x_n = \log y_n\rightarrow\xi=\log\eta .$
$\therefore (x_n') =\frac{x_1+x_2+\cdots+x_n}{n} = \log\sqrt[n]{y_1\cdots y_n}$
By using the above corollary 2.2.11.1, we get, $\log\sqrt[n]{y_1\cdots y_n}\rightarrow\log\eta \implies\sqrt[n]{y_1\cdots y_n}\rightarrow\eta$
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.
$\underline{\textit{Generalization of Cauchy’s Theorems}}$
Theorem 2.2.14: Let $(x_0,\cdots,x_n)$ be a null sequence and let the coefficients of the following system, satisfy the following two conditions :
- Every column contains a null sequence i.e for a fixed $p\geq0,\,a_{np}\rightarrow0$ as $n\rightarrow\infty$
- There exits a constant $K$ , such that, sum of the absolute values of terms in any one row i.e for every $n$, $|a_{n0}|+|a_{n1}|+\cdots+|a_{nn}|$
Then the sequence formed by numbers $x_n' = a_{n0}x_0 + a_{n1}x_1+\cdots+a_{nn}x_n$ is also a null sequence.
Proof: For any given $\epsilon>0,\,\exists\,m$ such that $|x_n|<\epsilon,\,\forall\,n>m$.
$\therefore |x_n'|<|a_{n0}x_0 + a_{n1}x_1+\cdots+a_{nm}x_m| + \frac{\epsilon}{2}$
From condition (1) we may now choose $n_0>m$ such that,
$|a_{n0}x_0 + a_{n1}x_1+\cdots+a_{nm}x_m|<\frac{\epsilon}{2},\,\forall\,n>n_0 \implies |x_n|<\epsilon,\forall\,n>n_0$
Q.E.D
Corollary 2.2.14.1: If our coefficients $a_{\lambda\mu}$ are substituted by any other numbers $a_{\lambda\mu}'=\alpha_{\mu}a_{\lambda\mu}$, obtained from from numbers $a_{\lambda\mu}$ by multiplication by factor $\alpha_{\lambda\mu}$, all in absolute values less than fixed $\alpha$, then the numbers $x_n''= a_{n0}'x_0+\cdots+a_{nn}'x_n$ also forms a null sequence.
Proof: The $a'_{\lambda\mu}$ satisfies conditions (1) and (2) in above theorem, hence for a fixed $p\geq 0$, $a_{np}' \rightarrow 0$, by theorem 2.1.1. Also $|a_{n0}'|+|a_{n1}'|+\cdots+|a_{nn}'|<\alpha K$. Hence applying theorem 2.1.2, we get the desired result.
Q.E.D
Theorem 2.2.15: If $(x_n)\rightarrow\xi$ and coefficient $a_{\mu\eta}$ beside the condition (1) and (2), satisfy, a new condition
c. $A_n = a_{n0}+\cdots+a_{nn} \rightarrow 1$,
then all also the numbers $x_n' = a_{n0}x_n+\cdots+a_{nn}x_n\rightarrow\xi$
Proof: Let $x_n' = A_n\xi + a_{nn_0}(x_0-\xi) +\cdots+a_{nn}(x_n-\xi)$
$(x_n-\xi)$is a null sequence and $(a_{nn})$ is a bounded sequence and $(A_n-1)$ is a null sequence, while $\xi$ is a constant. Then using theorems 2.1.1 and 2.1.2 $(x_n')$ 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 :
$(A) :$ Is a given sequence $(x_n)$ 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. $(B) :$ To what limit $\xi$, does the sequence $(x_n)$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 $\xi$ 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
$(B'):$ Two convergent sequences $(x_n)$and $(x_n')$ 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.
$\underline{\textit{First Main Criterion (For Monotone Sequences)}}$
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 $(x_n)$ is increasing and unbounded then $(x_n)\rightarrow\infty$ and similarly if $(y_n)$is decreasing and unbounded, $(y_n)\rightarrow-\infty$
b) If $(x_n)$ is increasing and bounded then $x_n\rightarrow\xi$ similar case for decreasing sequences.
Part A: If a sequence $(x_n)$ is monotonically increasing then $x_{n+1}>x_n,\,\forall\,n$. Hence the sequence is bounded below (or on the left) as $x_{n}\geq x_1,\,\forall\,n$.
Consider $(x_n)$is not bounded on right, i.e for any given $G>0$, we always find an index $n_0$ for which $x_{n_0}>G\implies x_n>G,\,\forall\,n>n_0$
Hence by definition 2.2.4, $x_n \rightarrow +\infty$
Similarly by some interchanges we can prove that a monotonically decreasing sequence $(y_n)$not bounded, $y_n\rightarrow-\infty$
Part B: Now let $(x_n)$be a monotonically increasing bounded sequence. i.e $\exists\, K$ such that $x_1\leq x_n \lt K, \forall n$
The interval $J_1=(x_1,K)$ contains all the terms of $(x_n)$. We will apply method of successive bisection to this interval and after bisecting the interval we denote right or left half of $J_1$as $J_2$depending on whether it does or doesn't contains points of $(x_n)$
We apply the same rules on $J_2$ and construct another interval $J_3$ and so on. Now for any interval $J_k$we can choose $n_k$ such that $\forall\,n>n_k$, all such $(x_n)$lies to right of $J_k$ and no such term of $(x_n)$ is to the left. Let $\xi$ be the number determined by the nest $(J_n)$.
Now, for any $\epsilon>0$, we can choose $p\in\mathbb{N}$such that the length of $J_p$ is less than $\epsilon$ .
$\therefore \forall \,n>n_p$, the $(x_n)$'s along with $\xi$ resides
$J_p \implies |x_n-\xi|<\epsilon,\,\forall\,n>n_p\implies (x_n-\xi)$ in is a null sequence i.e $x_n\rightarrow\xi$.
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,
$\underline{\textit{Second Main Criterion}}$
Theorem 2.3.2: An arbitrary sequence $(x_n)$is said to be Cauchy Sequence if and only if, given $\epsilon>0$, a natural number $n_0=n_0(\epsilon)$ can always be assigned such that for any two indices $n\,\&\,m$both greater than $n_0$, we have $|x_n-x_m|<\epsilon$.
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 $(x_n)$ is convergent and tends to $\xi$ then for any $\epsilon>0,\,\exists\,n_0$ such that $|x_n-\xi|<\frac{\epsilon}{2},\,\forall\,n>n_0$.
Now we choose, $n'>n_0$such that, $|x_{n'}-\xi|<\frac{\epsilon}{2}$.
$\therefore |x_n-x_{n'}|=|(x_n-\xi)-(x_{n'}-\xi)|\leq|x_n-\xi|+|x_{n'}-\xi|<\epsilon$
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 $\epsilon>0 ,$ we can always assign an index $p$ such that for all indices $n>p , |x_n-x_p|<\epsilon$.
Now to construct a nest of intervals let $\epsilon = \frac{1}{2},\frac{1}{4},\cdots,\frac{1}{2^k}$, we get
1) There is an index $p_1$ such that for $n>p_1 , |x_n-x_{p_1}|<\frac{1}{2}$
2) There exists an index $p_2$ such that for $n>p_2,\,|x_n-x_{p_2}|<\frac{1}{4}$
3) Going on like this, there exists $p_k$ such that $\forall n>p_k , |x_n-x_{p_k}|<\frac{1}{2^k}$
Accordingly we form intervals $J_k$such that intervals $(x_{p_1}-\frac{1}{2},x_{p_1}+\frac{1}{2})$ is called $J_1$. It contains all the $(x_n)$'s for which $n>p_1$. Hence it contains $x_{p_2}$
Therefore it also contains whole of interval $(x_{p_2}-\frac{1}{4},x_{p_2}-\frac{1}{4})$which contains all $(x_n)$'s such that $n>p_2$. $J_2$ is an interval formed by common points of $J_1$and this interval.
Going on like this, we construct ${J}_{k}$