Numerical sequences and series

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Chapter 3: Numerical sequences and series As the title indicates, this chapter will deal primarily with sequences and series of complex numbers. The basic facts about convergence, however, are just as easily explained in a more general setting. The first three sections will therefore be concerned with sequences in Euclidean spaces, or even in metric spaces.

Contents
Contents
 1.  Convergent sequences
 2.  Subsequences
 3.  Cauchy sequences
 4.  Upper and lower limits
 5.  Sone special sequences
 6.  Series
 7.  Series of nonnegative terms
 8.  The number e
 9.  The root and ratio tests
 10.  Power series
 11.  Summation by parts
 12.  Absolute converge
 13.  Addition and multiplication of series
 14.  Rearrangement

1. Convergent sequences

It might be well to point out that our definition of "convergent sequence" depends not only on \{p_n\} but also on X; for instance, the sequence \{1/n\} converges in \mathbb{R}^1 (to 0), but fails to converge in the set of all positive real numbers [with d(x, y) = | x- y |]. In cases of possible ambiguity, we can be more precise and specify "convergent in X" rather than "convergent."

We now summarize some important properties of convergent sequences in metric spaces.

For sequences in \mathbb{R}^k we can study the relation between convergence, on the one hand, and the algebraic operations on the other. We first consider sequences of complex numbers.

It is clear that \{p_n\} converges to p if and only if every subsequence of \{p_n\} converges to p.

Proof. Theorem theorem321. Let E be the range of \{p_n\}. It is trivial when E is finite. If E is infinite, Theorem theorem235 shows that E has a limit point p \in X. Choose n_1 so that d(P,P_{n_1})<1. Having chosen n_1,\dots , n_{i-1} we see from Theorem theorem222 that there is an integer n_1 > n_{i-1} such that d(p,p_{n_i})<1/i. Then \{p_{n_i}\} converges to p. (2) follows from (1), since Theorem theorem239 implies that every bounded subset of \mathbb{R}^k lies in a compact subset of \mathbb{R}^k.

Theorem theorem322. Let E^\ast be the set of all subsequential limits of \{p_n\} and let q be a limit point of E^\ast. We have to show that q \in E^\ast. Choose n_1 so that p_{n_1}\neq q. (If no such n_1 exists, then E^\ast has only one point, and there is nothing to prove.) Put \delta = d(q, p_{n_1}). Suppose n_1, \dots , n_{i-1} are chosen. Since q is a limit point of E^\ast, there is an x \in E^\ast with d(x, q)<2^{-i}\delta. Since x \in E^\ast, there is an n_i > n_{i-1} such that d(x,p_{n_i})<2^{-i}\delta. Thus d(q, p_{n_i})<2^{1-i}\delta for i = 1, 2, 3,\dots. This says that \{p_n\} converges to q. Hence q \in E^\ast.
3. Cauchy sequences If \{p_n\} is a sequence in X and if E_N consists of the points p_N, p_{N+ 1},p_{N+ 2} , \dots,
it is clear from the two preceding definitions that \{p_n\} is a Cauchy sequence if and only if
\[ \lim_{N\to\infty}\operatorname{diam}E_N=0.\]

The difference between the definition of convergence and the definition of a Cauchy sequence is that the limit is explicitly involved in the former, but not in the latter. Thus (2) may enable us to decide whether or not a given sequence converges without knowledge of the limit to which it may converge.

The fact that a sequence converges in \mathbb{R}^k if and only if it is a Cauchy sequence is usually called the Cauchy criterion for convergence.

Thus Theorem theorem332 says that all compact metric spaces and all Euclidean spaces are complete. Theorem theorem332 implies also that every closed subset E of a complete metric space X is complete. (Every Cauchy sequence in E is a Cauchy sequence in X, hence it converges to some p \in X, and actually p\in E since E is closed.) An example of a metric space which is not complete is the space of all rational numbers, with d(x, y) = | x- y|.

There is one important case in which convergence is equivalent to boundedness; this happens for monotonic sequences in \mathbb{R}^1.

Proof. Theorem theorem332. (2) Let \{p_n\} be a Cauchy sequence in the compact space X. For N = 1, 2, 3, \dots , let E_N be the set consisting of p_N, p_{N+l}, p_{N+2}, \dots. Then
\begin{equation}
\lim_{N\to\infty}\operatorname{diam}\bar E_N=0
\end{equation}
by Theorem theorem331(1). Being a closed subset of the compact space X, each \bar E_N is compact. Also E_N\supset E_{N+ 1}, so that \bar E_N\supset \bar E_{N+1}. Theorem theorem331(2) shows now that there is a unique p \in X which lies in every \bar E_N. Let \varepsilon>0 be given. By (31) there is an integer N_0 such that \operatorname{diam}\bar E_N<\varepsilon if N \geqslant N_0. Since p\in \bar E_N, it follows that d(p, q)<\varepsilon for every q \in\bar E_N, hence for every q \in E_N. In other words, d(p, p_n)<\varepsilon if n\geqslant N_0. This says precisely that p_n\to p.

(3) Let \{\mathbf{x} _ n\} be a Cauchy sequence in \mathbb{R}^k. Define E_N as in (2), with \mathbf{x} _ i in place of p_i. For some N, \operatorname{diam} E_N< 1. The range of \{\mathbf{x} _ n\} is the union of E_N and the finite set \{\mathbf{x} _ 1,\dots,\mathbf{x} _ {N-1}\}. Hence \{\mathbf{x} _ n\} is bounded. Since every bounded subset of \mathbb{R}^k has compact closure in \mathbb{R}^k (Theorem theorem239), (3) follows from (2).
4. Upper and lower limits

We close this section with a theorem which is useful, and whose proof is quite trivial:

We shall compute the limits of some sequences which occur frequently.

In the remainder of this chapter, all sequences and series under consideration will be complex-valued, unless the contrary is explicitly stated.

It is clear that every theorem about sequences can be stated in terms of series (putting a_1 = s_1, and a_n = s_n - s_{n - 1} for n > 1), and vice versa. But it is nevertheless useful to consider both concepts.

The Cauchy criterion (Theorem theorem332) can be restated in the following form:

In particular, by taking m = n, we obtain |a_n|<\varepsilon\, (n\geqslant N). In other words:

The condition a_n\to0 is not, however, sufficient to ensure convergence of \sum a_n.

Theorem theorem333, concerning monotonic sequences, also has an immediate counterpart for series.

We now turn to a convergence test of a different nature, the so-called "comparison test."

The comparison test is a very useful one; to use it efficiently, we have to become familiar with a number of series of nonnegative terms whose convergence or divergence is known.
7. Series of nonnegative terms The simplest of all is perhaps the geometric series.

In many cases which occur in applications, the terms of the series decrease monotonically. The following theorem of Cauchy is therefore of particular interest. The striking feature of the theorem is that a rather "thin" subsequence of \{a_n\} determines the convergence or divergence of \sum a_n.

As a further application of Theorem theorem372, we have:

We do not wish to go any deeper into this aspect of convergence theory, and refer the reader to other materials.
8. The number e

Since the partial sum is bounded (\leqslant 3), the series converges, and the definition makes sense. In fact, the series converges very rapidly and allows us to compute e with great accuracy.

It is of interest to note that e can also be defined by means of another limit process:

The rapidity with which the series \sum\frac{1}{n!} converges can be estimated as follows:
\[0<e-s_n<\frac{1}{n!n}.\]

Thus s_{10}, for instance, approximates e with an error less than 10^{-7}. The inequality above is of theoretical interest as well, since it enables us to prove the irrationality of e very easily.

Actually, e is not even an algebraic number.
9. The root and ratio tests

Note: The knowledge that \lim a_{n+1}/a_n = 1 implies nothing about the convergence of \sum a_n. The series \sum 1/n and \sum1/n^2 demonstrate this.

The ratio test is frequently easier to apply than the root test, since it is usually easier to compute ratios than nth roots. However, the root test has wider scope. More precisely: Whenever the ratio test shows convergence, the root test does too; whenever the root test is inconclusive, the ratio test is too. This is a consequence of Theorem theorem393.

Neither of the two tests is subtle with regard to divergence. Both deduce divergence from the fact that a_n does not tend to zero as n \to\infty.

In general, the series will converge or diverge, depending on the choice of z. More specifically, with every power series there is associated a circle, the circle of convergence, such that (32) converges if z is in the interior of the circle and diverges if z is in the exterior (to cover all cases, we have to consider the plane as the interior of a circle of infinite radius, and a point as a circle of radius zero). The behavior on the circle of convergence is much more varied and cannot be described so simply.

Formula 33, the so-called "partial summation formula," is useful in the investigation of series of the form \sum a_nb_n, particularly when \{b_n\} is monotonic. We shall now give applications.

Series for which (2) holds are called "alternating series"; the theorem was known to Leibnitz.

For series of positive terms, absolute convergence is the same as convergence.

If \sum a_n converges, but \sum|a_n| diverges, we say that \sum a_n converges non-absolutely.

The comparison test, as well as the root and ratio tests, is really a test for absolute convergence, and therefore cannot give any information about nonabsolutely convergent series. Summation by parts can sometimes be used to
handle the latter. In particular, power series converge absolutely in the interior of the circle of convergence.

We shall see that we may operate with absolutely convergent series very much as with finite sums. We may multiply them term by term and we may change the order in which the additions are carried out, without affecting the sum of the series. But for nonabsolutely convergent series this is no longer true, and more care has to be taken when dealing with them.
13. Addition and multiplication of series

Thus two convergent series may be added term by term, and the resulting series converges to the sum of the two series. The situation becomes more complicated when we consider multiplication of two series. To begin with, we have to define the product. This can be done in several ways, we shall consider the so-called "Cauchy product."

This definition may be motivated as follows. If we take two power series \sum a_nz^n and \sum b_nz^n, multiply them term by term, and collect terms containing the same power of z, we get
\[\sum_{n=0}^{\infty}a_nz^n\sum_{n=0}^{\infty}b_nz^n=a_0b_0+(a_0b_1+a_1b_0)z+\dots=c_0+c_1z+c_2z^2+\dots.\]
Setting z=1, we arrive at the above definition.

{Example} If A_n=\sum_{k=0}^{n}a_k, B_n=\sum_{k=0}^{n}b_k, C_n=\sum_{k=0}^{n}c_k, and A_n\to A, B_n\to B, then it is not at all clear that \{C_n\} will converge to AB, since we do not have C_n=A_nB_n. The dependence of \{C_n\} on \{A_n\} and \{B_n\} is quite a complicated one. We shall now show that the product of two convergent series may actually diverge.

The series \sum (-1)^n/(\sqrt{n+1}) converges. We form the product of itself and obtain a diverge series.

In view of the next theorem, due to Mertens, we note that we have here considered the product of two nonabsolutely convergent series.

To prove this, put \beta_n=B_n-B. Then C_n=A_nB+a_0\beta_n+a_1\beta_{n-1}+\dots+a_n\beta_0. Put \gamma=a_0\beta_n+\dots+a_n\beta_0. It suffices to show that \lim \gamma_n=0.

Put \alpha=\sum|a_n|. Let \varepsilon>0 be given. We can choose N such that |\beta_n|<\varepsilon for n\geqslant N, in which case
\[
|\gamma_n|\leqslant|\beta_0a_n+\dots+\beta_Na_{n-N}|+|\beta_{N+1}a_{n-N-1}+\dots+\beta_na_0| \leqslant|\beta_0a_n+\dots+\beta_Na_{n-N}|+\varepsilon\alpha.
\]
Keeping N fixed, and letting n\to\infty, we get \limsup|\gamma_n|\leqslant\varepsilon\alpha. Since \varepsilon is arbitrary, \gamma\to0.

Another question which may be asked is whether the series \sum c_n, if convergent, must have the sum AB. Abel showed that the answer is in the affirmative.

Here no assumption is made concerning absolute convergence. We shall give a simple proof (which depends on the continuity of power series) in chapter 8.
14. Rearrangement Let \sum a^\prime_n be a rearrangement of \sum a_n.

If \{s _ n\}, \{s^\prime _ n\} are the sequences of partial sums of \sum a_n, \sum a^\prime_n, it is easily seen that, in general, these two sequences consist of entirely different numbers. We are thus led to the problem of determining under what conditions all rearrangements of a convergent series will converge and whether the sums are necessarily the same.

Proof. Theorem theorem3141. Let p_n=a_n^+, q_n=a_n^-. Then p_n-q_n=a_n, q_n+q_n=|a_n|, p_n\geqslant0, q_n\geqslant0. Note that
\[\sum_{n=1}^{\infty}p_n=+\infty,\quad\sum_{n=1}^{\infty}q_n=+\infty.\]
Now let P_1,P_2,\dots denote the nonnegative terms of \sum a_n, in the order in which they occur, and let Q_1,Q_2,\dots be the absolute values of the negative terms of \sum a_n, also in their original order.

The series \sum P_n,\sum Q_n differ from \sum p_n,\sum q_n only by zero terms, and are therefore divergent.

We shall construct sequences \{m_n\},\{k_n\}, such that the series
\begin{equation}
P_1+\dots+P_{m_1}-Q_1-\dots-Q_{k_1}+P_{m_1+1}+\dots+P_{m_2}-Q_{k_1+1}-\dots-Q_{k_2}+\dots,
\end{equation}
which clearly is a rearrangement of \sum a_n, satisfies the liminf of the partial sum is \alpha and limsup is \beta.

Choose real-valued sequences \{\alpha_n\},\{\beta_n\} such that \alpha_n\to\alpha, \beta_n\to\beta, \alpha_n<\beta_n, \beta_1>0. Let m_1,k_1 be the smallest integers such that
\[P_1+\dots+P_{m_1}>\beta_1,\quad P_1+\dots+P_{m_1}-Q_1-\dots-Q_{k_1}<\alpha_1;\]
let m_2,k_2 be the smallest integers such that
\[\begin{gathered}
P_1+\dots+P_{m_1}-Q_1-\dots-Q_{k_1}+P_{m_1+1}+\dots+P_{m_2}>\beta_2, \
P_1+\dots+P_{m_1}-Q_1-\dots-Q_{k_1}+P_{m_1+1}+\dots+P_{m_2}-Q_{k_1+1}-\dots-Q_{k_2}<\alpha_2;
\end{gathered}\]
and continue in this way. This is possible since \sum P_n and \sum Q_n diverge.

If x_n,y_n denote the partial sums of (35) whose last terms are P_{m_n},Q_{k_n}, then |x_n-\beta_n|<P_{m_n}, |y_n-\alpha_n|<Q_{k_n}. Since P_n\to0 and Q_n\to0 as n\to\infty, we see that x_n\to\beta, y_n\to\alpha.

Finally, it is clear that no number less than \alpha or greater than \beta can be a subsequential limit of the partial sums of (35).

Theorem theorem3142. Let \sum a^\prime_n=\sum a_{k_n} be a rearrangement, with partial sums s^\prime_n. Given \varepsilon>0, there exists an integer N such that m\geqslant n\geqslant N implies \sum_{i=n}^{m}|a_i|<\varepsilon. Now choose p such that the integers 1,2,\dots,N are all contained in the set k_1,\dots,k_p. Then if n>p, the numbers a_1,\dots,a_N will cancel in the difference s_n-s'_n, so that |s_n-s^\prime_n|\leqslant\varepsilon. Hence \{s^\prime_n\} converges to the same sum as \{s_n\}.