Infinite series - theory and applications
The Series ∑∞n=1(1n)
Keywords: tests of convergence, limiting value, power series, rate of convergence, transformation of series, numerical valuations
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 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 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 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 corresponds to a definite real number then the numbers are said to form a sequence.
Definition 2.1.2 : (Bounded Sequence ) - A sequence is said to be bounded if and only if there exists a constant number such that
Definition 2.1.3 : (Monotonic Sequence) - A sequence is said to be monotonically increasing if and is monotonically decreasing if .
Definition 2.1.3 : (Null Sequence) - A sequence is said to be a null sequence if for every , there exists a number such that, .
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 , the -neighbourhood of contains infinitely many of terms of , then 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 is a null sequence and the terms of the sequence , for every beyond a certain , satisfies the condition or more generally , where is a arbitrary fixed positive number, then is also a null sequence.
Proof : The proof can be quite easily laid down.
We know that for every such that .
Now we can choose , such that for all where .
But for all these values of is a null sequence.
Theorem 2.1.2: If is a null sequence and is any arbitrary bounded sequence then also forms a null sequence.
Proof: Let such that . We can now choose a number such that, for every .
Now . Hence for every is a null sequence.
Theorem 2.1.3: If is a null sequence, then every sub-sequence of is a null sequence.
Proof : Let is a subsequence of .
is some positive integer.
Now, for any such that then for any such , since , certainly, if .
Theorem 2.1.4: Let an arbitrary sequence be separated into two sub-sequences and , so that, every term of belongs to one and only one of these sub-sequences. If and are both null sequences, then so is itself.
Proof : For a any given , such that and
Now all the elements of and will be arranged in a definite order in i.e they would have definite indices and let the terms with indices , now has indices in .
If is a positive integer then .
Theorem 2.1.5: If is a null sequence and is an arbitrary rearrangement of it, then is also a null sequence.
Proof: For any such that
From the indices of obtained after rearrangement, let be
Then it's logical to see that for every is also a null sequence.
Theorem 2.1.6: If is a null sequence and is obtained by any finite number of
alterations, then is also a null sequence.
Proof: After doing some finitely many alterations to , we have after some onwards,
If every , for has not changed in and has received index then there exists such that for any given .
Theorem 2.1.7: If and are two null sequences and if the sequence is so related to them by
Now, as we understand the null sequences better, let's have a look at the algebra of null sequences.
Theorem 2.1.8: If and are two null sequences then is also a null sequence. (This theorem briefly means, two null sequences can be added together.)
Proof : For any arbitrary such that,
Now, let be a number such that ,
hence is a null sequence.
Theorem 2.1.9: If and are two null sequences then is also a null sequence. (Similarly, this theorem briefly means, two null sequences can be subtracted together.)
Proof: We know that if is a null sequence and let be any constant real number then, is also a null sequence.
Hence, choosing , we get is also a null sequence.
Then it follows from above theorem, that is also a null sequence.
Similarly, the two null sequences can be multiplied term-wise. But division is not that straight forward. To check, consider where is non-zero for all , is not a null sequence. Also if is a null sequence, nothing can be said about.
Theorem 2.1.10: If , a sequence of absolute values of terms of , has a positive lower bound i.e a number exists, such that, for every , then the sequence of reciprocals, , is bounded.
Proof : One can easily see that if then for
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 . 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 is a given sequence, and if it is related to a definite number in such a way that forms a null sequence, then we say that the sequence converges to , or that it is convergent. The number is called the limiting value or limit of this sequence; the sequence is also said to converge to and we say that it's terms approach the (limiting) value , tend to , have the limit . This fact is expressed by the symbols as or .
Or in other words, which convey the same meaning,
Definition 2.2.2 : A sequence is said to converge to , or is said to be a limit of , if for every there exists a natural number such that for all , the terms satisfy, .
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 has a property that for an arbitrary positive number , there exists another number such that for every . Then we say that diverges to , tends to or is definitely divergent with limit . We write or or
By merely interchanging right and left we get,
Definition 2.2.6: A sequence 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.
Now, let as , but it converges indefinitely to zero.
One important notation that will be useful further.
Definition 2.2.7: If two sequence and , not necessarily convergent, are related to one another such that the quotient tends to a definite finite limiting value (limit) for , different from zero, and we write .
In particular, if this limit is , then we say that the sequences are asymptotically equal write as .
For example :
Theorem 2.2.1: A convergent sequence determines its limit quite uniquely.
Proof: If and , simultaneously, then and both are null sequences and from theorem 2.1.9 is also a null sequence.
Theorem 2.2.2: A convergent sequence is invariably, bounded and if then for limit .
[Every convergent sequence is a bounded sequence.]
Corollary 220.127.116.11: If then
Proof: Now, , hence by comparison test, .
Theorem 2.2.3 : If a convergent sequence has all it's terms different from zero and if it's limit then is bounded or in other words, a number ,exists such that for every , the numbers possesses a positive lower bound.
Theorem 2.2.4: If is a subsequence of then
Theorem 2.2.5: If a sequence can be divided into two sub-sequences of which each converges to then also converges to .
Theorem 2.2.6: If is an arbitrary rearrangement of then
Theorem 2.2.7: If and if is obtained by finite alterations of , then
Theorem 2.2.8: If and and if sequence lies between and from some onwards i.e , then
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.
Theorem 2.2.9: If and always implies and the statement holds for term by term addition for any fixed number - say P - of convergent sequences.
Proof: As and are null sequences, from theorem 2.1.8, we know that is also a null sequences.
Corollary 18.104.22.168: and always implies
Proof follows from theorem 2.1.9
Theorem 2.2.10: and always implies and the statement holds for term by term multiplication for any fixed number - say P - of convergent sequences.
Proof: We have . And now it follows from theorem 2.1.2, that forms a null sequence.
Theorem 2.2.11: and always implies, if every and also then
Proof: We have
Now from above theorem 2.2.10, the numerator is a null sequence and from theorem 2.2.3, is a bounded sequence. Hence finally using theorem 2.1.2, our theorem is proved.
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 is a null sequence, then the arithmetic mean is also a null sequence.
Proof: For any given such that .
For these 's, we have (Every term is )
The first fraction above is formed of finitely many terms, hence we can choose an such that for every is a null sequence.
Corollary 22.214.171.124: If then so does the arithmetic mean,
Proof: We have .
From the above theorem is a null sequence when is i.e
Now a similar theorem for the Geometric Mean.
Theorem 2.2.13: Let and the terms in it and be positive. Then the Geometric Mean i.e is a null sequence.
Now using properties of we get
By using the above corollary 126.96.36.199, we get,
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.
Theorem 2.2.14: Let 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 as
- There exits a constant , such that, sum of the absolute values of terms in any one row i.e for every ,
Then the sequence formed by numbers is also a null sequence.
Proof: For any given such that .
From condition (1) we may now choose such that,
Corollary 188.8.131.52: If our coefficients are substituted by any other numbers , obtained from from numbers by multiplication by factor , all in absolute values less than fixed , then the numbers also forms a null sequence.
Proof: The satisfies conditions (1) and (2) in above theorem, hence for a fixed , , by theorem 2.1.1. Also . Hence applying theorem 2.1.2, we get the desired result.
Theorem 2.2.15: If and coefficient beside the condition (1) and (2), satisfy, a new condition
then all also the numbers
is a null sequence and is a bounded sequence and is a null sequence, while is a constant. Then using theorems 2.1.1 and 2.1.2 is also a null sequence.
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 :
Is a given sequence 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. To what limit , does the sequence 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 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
Two convergent sequences and 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.
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 is increasing and unbounded then and similarly if is decreasing and unbounded,
b) If is increasing and bounded then similar case for decreasing sequences.
Part A: If a sequence is monotonically increasing then . Hence the sequence is bounded below (or on the left) as .
Consider is not bounded on right, i.e for any given , we always find an index for which
Hence by definition 2.2.4,
Similarly by some interchanges we can prove that a monotonically decreasing sequence not bounded,
Part B: Now let be a monotonically increasing bounded sequence. i.e such that
The interval contains all the terms of . We will apply method of successive bisection to this interval and after bisecting the interval we denote right or left half of as depending on whether it does or doesn't contains points of
We apply the same rules on and construct another interval and so on. Now for any interval we can choose such that , all such lies to right of and no such term of is to the left. Let be the number determined by the nest .
Now, for any , we can choose such that the length of is less than .
, the 's along with resides
in is a null sequence i.e .
By making, suitable changes we can prove the same for a bounded monotonically decreasing sequence.
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,
Theorem 2.3.2: An arbitrary sequence is said to be Cauchy Sequence if and only if, given , a natural number can always be assigned such that for any two indices both greater than , we have .
Every Cauchy sequence is convergent and every convergent sequence is a Cauchy Sequence.
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 is convergent and tends to then for any such that .
Now we choose, such that, .
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 we can always assign an index such that for all indices .
Now to construct a nest of intervals let , we get
1) There is an index such that for
2) There exists an index such that for
3) Going on like this, there exists such that
Accordingly we form intervals such that intervals is called . It contains all the 's for which . Hence it contains
Therefore it also contains whole of interval which contains all 's such that . is an interval formed by common points of and this interval.
Going on like this, we construct