Acquainting with Convergent Series

We began our study of convergent series with several poignant examples and a useful convergence criteria known as the Cauchy criterion. In particular, the geometric series comes into exceptional clutch.

Throughout this post, let $\mathbb K$ be an ordered field.

Theorem 1. For any real number $r$ , the geometric series

$\displaystyle \sum_{k=1}^\infty r^{k-1} = 1 + r + r^2 + \cdots$

converges if and only if $|r| < 1$ .

Since the convergence of series are intimately connected to the convergence of series, it should come to no surprise that most convergence properties in the latter port over to the former. We restate them in the context of series.

Theorem 2. Let $\{x_n\}$ , $\{y_n\}$ be $\mathbb K$ -sequences and $c \in \mathbb K$ . Suppose $\displaystyle \sum_{k=1}^\infty x_n$ and $\displaystyle \sum_{k=1}^\infty y_n$ converge. Then

$\displaystyle \sum_{k=1}^\infty (x_n + y_n) = \sum_{k=1}^\infty x_n + \sum_{k=1}^\infty y_n,\quad \sum_{k=1}^\infty (cx_n) = c \sum_{k=1}^\infty x_n.$

In particular, the general geometric series

$\displaystyle \sum_{k=1}^\infty ar^{k-1} = a + ar + ar^2 + \cdots = \frac{a}{1-r}$

if and only if $|r| < 1$ .

Unfortunately, more care needs to be taken to multiply two series together, since in general $+$ and $\times$ refer to different calculations.

Another important observation is that the convergence of series is a lot more dependent on its tail behaviour than its whole behaviour.

Theorem 3. Let $\{x_n\}$ be a $\mathbb K$ -sequence. Then $\{s_n\}$ converges if and only if there exists $N \in \mathbb N$ such that $\{s_{n+N-1}\}$ converges. Symbolically,

$\displaystyle \sum_{k=1}^\infty x_n\ \text{converges} \iff \exists N \in \mathbb N:\sum_{k=N+1}^\infty x_k = x_N + x_{N+1} + \cdots \ \text{converges}.$

Proof. We observe that for $n > N$ , $s_n = s_N + (s_n - s_N)$ so that $s_n \to s$ converges if and only if $s_{n+N-1} \to s - s_N$ .

If, however, we instead let the starting value $N$ be the term that increases, we have the nth-term divergence test. This does not help us establish convergence by any means, but it sure helps us know when a series does not converge.

Theorem 4. Let $\{x_n\}$ be a $\mathbb K$ -sequence.

If $s_\infty$ converges, then $x_n \to 0$ .
Equivalently, if $x_n \not\to 0$ , then $s_\infty$ diverges.

Proof. If $s_n \to s \in \mathbb K$ , then $x_n = s_{n+1} - s_n \to s - s = 0$ .

Example 1. For the sequence $\{x_n\} = \{1\}$ , since $x_n = 1 \not\to 0$ , $s_\infty$ diverges.

Unfortunately, the converse is not true. We can have sequences $\{x_n\}$ with $x_n \to 0$ and yet $s_\infty$ diverges. The crucial object of interest here is the harmonic series.

Theorem 5. The harmonic series $1 + 1/2 + 1/3 + 1/4 + \cdots$ diverges.

Proof. Write

$\displaystyle s_n = 1 + \frac 12 + \frac 13 + \frac 14 + \cdots + \frac 1n.$

By observation,

$\begin{aligned} s_{2n} &= 1 + \frac 12 + \frac 13 + \frac 14 + \cdots + \frac 1{2n-1} + \frac 1{2n}\\ &\geq 1 + \frac 12 + \frac 14 + \frac 14 + \cdots + \frac 1{2n} + \frac 1{2n} \\ &= \frac12 + 1 + \frac 12 + \frac 13 + \cdots + \frac 1n \\ &= \frac 12 + s_n. \end{aligned}$

Suppose for a contradiction that $s_\infty = s \in \mathbb K$ . Then taking $n \to \infty$ on both sides yields

$\displaystyle s \geq \frac 12 + s \quad \Rightarrow \quad 0 \geq 1 > 0,$

a contradiction.

Using the various convergent series we know, we can use convergence tests to prove that other series converge. We have many more convergence tests to discuss, but for now we will give a sneak peek into the first convergence test of interest. This is called the comparison test.

Theorem 5. Call an $\mathbb R$ -sequence $\{x_n\}$ non-negative if $x_n \geq 0$ for any $n$ . Let the non-negative sequences $\{x_n\},\{y_n\}$ have partial sums $\{s_n\},\{t_n\}$ .

Suppose for any $n \in \mathbb N$ , $0 \leq x_n \leq y_n$ .

If $t_\infty$ converges, so does $s_\infty$ , and $s_\infty \leq t_\infty$ .
Equivalently, if $s_\infty$ diverges, so does $t_\infty$ .

Proof. Since each $0 \leq x_n \leq y_n$ , it is clear that $\{s_n\}$ and $\{t_n\}$ are non-decreasing. If $t_\infty \to t$ , then $s_n \leq t_n \leq t$ so that $\{s_n\}$ is bounded above. By the monotone convergence theorem, $s_n \to s_\infty$ for some $s_\infty \in \mathbb R$ . Furthermore, since $s_n \leq t_n$ , taking $n \to \infty$ yields $s_\infty \leq t_\infty$ .

By Theorem 3, we only need these properties to be true for sufficiently large $N$ for the convergence test to still be valid. For clarity, we will use this property without any further elaboration.

The next post will be relevant to provide a slew of convergence tests to help students in their various computational advanced calculus courses.

—Joel Kindiak, 28 Dec 24, 1326H

KindiakMath

Acquainting with Convergent Series

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Acquainting with Convergent Series

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