📂Calculus

A Comprehensive Summary of Various Series Tests in Analysis

Overview

This post will introduce several series convergence tests without diving into their proofs. It is often more valuable to utilize these tests as facts, especially since the proofs can be quite tedious.

In this post, we use the following notations:

• $\mathbb{N}$ is the set containing all natural numbers.
• $\mathbb{R}$ is the set containing all real numbers, and $\overline{\mathbb{R}}$ is the extended real number set that includes $\pm \infty$.
• $\left\{ a_{k} \right\}_{k \in \mathbb{N}}, \left\{ b_{k} \right\}_{k \in \mathbb{N}} \subset \mathbb{R}$ is a sequence of real numbers.
• $\exists \lim_{k \to \infty} x_{k}$ means that the limit of $x_{k}$ exists in $\mathbb{R}$, meaning it converges. Conversely, $\not\exists \lim_{k \to \infty} x_{k}$ means that the limit of $x_{k}$ does not exist in $\mathbb{R}$, meaning it diverges.
• For sufficiently large $k$, when $\displaystyle \lim_{k \to \infty} { {a_k} \over {b_k} } = 1$ then it is expressed as $a_k \approx b_k$.
• $b_k \downarrow 0$ means that $b_{k}$ is a decreasing sequence and converges to $0$ while always taking a value greater than or equal to $0$.

Real Sequences ¹

Divergence Test

If $\lim _{ k \to \infty }{ { a }_{ k }} \ne 0$ then $\sum _{ k =1 }^{ \infty }{ { a }_{ k }}$ diverges: $$\lim _{ k \to \infty }{ { a }_{ k }} \ne 0 \implies \not\exists \sum _{ k =1 }^{ \infty }{ { a }_{ k }}$$

• The Divergence Test is the only method you might encounter in high school. It’s a pity that it cannot determine convergence itself, but it’s the easiest and fastest when showing divergence.

Cauchy’s Criterion

The convergence of $\sum _{ n=1 }^{ \infty }{ { a }_{ n }}$ is equivalent to $\lim_{n \to \infty} \sum _{ k=n }^{ n+m }{ { a }_{ k }}=0$: $$\exists \sum_{k=1}^{\infty} a_{k} \iff \left( \forall \varepsilon > 0 , \exists N \in \mathbb{N} : m \ge n \ge N \implies \left| \sum_{k=n}^{m} a_{k} \right| < \varepsilon \right)$$

• This is called the Cauchy Criterion because the theorem utilizes the fact that, in the space of real numbers $\mathbb{R}$, a converging sequence being a Cauchy sequence is equivalent.

Nonnegative Sequences ²

This section deals with sequences where each term is greater than or equal to $0$, denoted $a_{k} \ge 0$.

Integral Test

Let’s assume the decreasing function $f: [1,\infty) \to \mathbb{R}$ is always greater than $0$. The convergence of $\sum _{ k =1 }^{ \infty }{ { f }( k )}$ is equivalent to $\int_{1}^{\infty} f(x) dx < \infty$: $$\exists \sum_{k=1}^{\infty} f(k) \iff \int_{1}^{\infty} f(x) dx < \infty$$

• The Integral Test is proven using the fact that $f(n+1) \le \int_{n}^{n+1} f(x) dx \le f(n)$. It’s one of those rare criteria where the proof itself is interesting.

$p$-Series Test

The convergence of $\sum _{ k=1 }^{ \infty } k^{-p}$ is equivalent to $p>1$: $$\exists \sum_{k=1}^{\infty} {{ 1 } \over { k^{p} }} \iff p > 1$$

• The $p$-Series Test essentially means that if you increase the exponent even slightly in a harmonic series, it converges; otherwise, it diverges. It’s a corollary derived by inserting a geometric series into the integral test, but because it’s so simple and useful, it’s used even more often than the integral test.

Comparison Test

For sufficiently large $k$, let’s say $0 \le a_k \le b_k$. If $\sum _{ k=1 }^{ \infty }{ { b }_{ k }}$ converges, then $\sum _{ k=1 }^{ \infty }{ { a }_{ k }}$ also converges: $$\begin{align*} \sum_{k=1}^{\infty} b_{k} < \infty \implies & \sum_{k=1}^{\infty} a_{k} < \infty \\ \sum_{k=1}^{\infty} a_{k} = \infty \implies & \sum_{k=1}^{\infty} b_{k} = \infty \end{align*}$$

• The Comparison Test compares another series known to converge to determine the convergence as its name suggests. Using the contrapositive, you can similarly check if a series diverges.

Limit Comparison Test

For sufficiently large $k$, let’s say $a_k \ge 0$ and $b_k>0$. $L := \lim_{k \to \infty} { {a_k} \over {b_k} } \in \overline{\mathbb{R}}$ is some extended real number. (1) If $0<L<\infty$ then both $\sum _{ n=1 }^{ \infty }{ { a }_{ n }}$ and $\sum _{ n=1 }^{ \infty }{ { b }_{ n }}$ either converge or diverge together. (2) If $L=0$ and $\sum _{ n=1 }^{ \infty }{ { b }_{ n }}$ converges, then $\sum _{ n=1 }^{ \infty }{ { a }_{ n }}$ also converges. (3) If $L=\infty$ and $\sum _{ n=1 }^{ \infty }{ { b }_{ n }}$ diverges, then $\sum _{ n=1 }^{ \infty }{ { a }_{ n }}$ also diverges: $$\begin{align*} 0 < L < \infty \implies & \left( \exists \sum_{k=1}^{n} a_{k} \iff \exists \sum_{k=1}^{n} b_{k} \right) \\ L = 0 \implies & \left( \exists \sum_{k=1}^{n} b_{k} \implies \exists \sum_{k=1}^{n} a_{k} \right) \\ L = \infty \implies & \left( \not\exists \sum_{k=1}^{n} b_{k} \implies \not\exists \sum_{k=1}^{n} a_{k} \right) \end{align*}$$

• The Limit Comparison Test is like the comparison test but for when it’s difficult to show the original series converges by comparing it with another converging series. Though the conditions might seem strenuous, they’re often easy to satisfy when simply proving convergence, making this very useful.

Absolute Convergence ³

For an infinite series $S = \sum_{k=1}^{\infty} a_{k}$, if $\sum_{k=1}^{\infty} \left| a_{k} \right|$ converges, $S$ is defined to converge absolutely. Accordingly, a series that doesn’t converge absolutely but does converge on its own is said to converge conditionally.

Root Test

For the limit supremum $r = \limsup_{k \to \infty} {{|a_k|} ^ {1 / k}}$ of $\left\{ \left| a_{k} \right|^{1/k} \right\}$, if $r<1$ then $\sum _{ n=1 }^{ \infty }{ { a }_{ k }}$ absolutely converges, if $r>1$ then $\sum _{ n=1 }^{ \infty }{ { a }_{ k }}$ diverges: $$\begin{align*} r < 1 \implies & \exists \sum_{k=1}^{\infty} \left| a_{k} \right| \\ r > 1 \implies & \not\exists \sum_{k=1}^{\infty} a_{k} \end{align*}$$

Ratio Test

Assuming $a_{k} \ne 0$, and $r = \lim_{k \to \infty} { {|a_{k+1}|} \over {|a_{k}|} } \in \overline{\mathbb{R}}$ is an extended real number. If $r<1$ then $\sum _{ k=1 }^{ \infty }{ { a }_{ k }}$ absolutely converges, if $r>1$ then $\sum _{ k=1 }^{ \infty }{ { a }_{ k }}$ diverges: $$\begin{align*} r < 1 \implies & \exists \sum_{k=1}^{\infty} \left| a_{k} \right| \\ r > 1 \implies & \not\exists \sum_{k=1}^{\infty} a_{k} \end{align*}$$

• The Root Test and Ratio Test have slightly stringent conditions but can outright prove absolute convergence, hence their popularity. Meanwhile, if $r=1$, methods like Dirichlet’s Test or Alternating Series Test can be used. The alternating series test can be directly derived from Dirichlet’s test, and if you’re only looking to establish convergence, considering an alternating harmonic series as an example is helpful.

Dirichlet’s Test

If the partial sum $s_n = \sum_{k=1}^{n} a_k$ is bounded and $k \to \infty$ when $b_k \downarrow 0$, then $\sum _{ k=1 }^{ \infty }{ { a }_{ k } {b}_{k}}$ converges: $$\left| s_n \right| < \infty , b_k \downarrow 0 \implies \exists \sum _{ k=1 }^{ \infty }{ { a }_{ k } {b}_{k}}$$

Alternating Series Test

If $k \to \infty$ when $b_k \downarrow 0$, then $\sum _{ n=1 }^{ \infty }{ (-1)^{k} {b}_{k}}$ converges. $$b_k \downarrow 0 \implies \exists \sum _{ k=1 }^{ \infty }{ (-1)^{-k} {b}_{k}}$$