Series and Sequences: Convergence Tests Explained

A sequence is an ordered list of numbers: $a_1,a_2,a_3,\dots$ A series is the sum of the terms of a sequence: $a_1+a_2+a_3+\cdots$. Surprisingly, some infinite series sum to finite values. Determining which ones do — and computing their sums — is the subject of this article.

This article covers what sequences and series are, the formal definition of convergence, the geometric series (the most important example), the major convergence tests with worked examples, power series and Taylor expansion, and the common pitfalls.

Sequences

A sequence is a function from positive integers to real numbers: $n\mapsto a_n$. Notation: $\{a_n\}$ or $(a_n{)}_{n=1}^{\infty }$.

Examples:

• $a_n=1/n$: $1,1/2,1/3,1/4,\dots$
• $a_n=(-1{)}^n$: $-1,1,-1,1,\dots$
• $a_n=n^2$: $1,4,9,16,\dots$

Convergence of a sequence. A sequence $\{a_n\}$ converges to a limit $L$ if ${\lim }_{n\to \infty }{a}_n=L$. Otherwise it diverges.

The first sequence above converges to 0. The third diverges to infinity. The second neither converges nor diverges to infinity — it oscillates.

Series

A series is the sum of a sequence's terms. We define the partial sum:

$S_n=a_1+a_2+\cdots +a_n={\sum }_{k=1}^n{a}_k$

The series ${\sum }_{n=1}^{\infty }{a}_n$ converges if the sequence of partial sums $\{S_n\}$ converges. Otherwise it diverges.

Critical distinction. A sequence and a series are different. The sequence $\{1/n\}$ converges to 0. The series $\sum 1/n$ diverges to infinity (this is the harmonic series, a classic).

The geometric series

The most important series in mathematics. For $|r|<1$:

${\sum }_{n=0}^{\infty }{r}^n=1+r+r^2+r^3+\cdots =\frac{1}{1-r}$

For $|r|\ge 1$, the series diverges.

Proof sketch. The partial sum is $S_n=1+r+\cdots +r^{n-1}$. Multiply by $r$: $r{S}_n=r+r^2+\cdots +r^n$. Subtract: $S_n-r{S}_n=1-r^n$, so $S_n=\frac{1-r^n}{1-r}$. As $n\to \infty$, if $|r|<1$, $r^n\to 0$, and $S_n\to \frac{1}{1-r}$.

Example 1: ${\sum }_{n=0}^{\infty }\frac{1}{2^n}=1+\frac{1}{2}+\frac{1}{4}+\cdots =\frac{1}{1-1/2}=2$.

The geometric series appears in compound interest, present-value calculations, probability problems (waiting times, geometric distribution), and many physics applications.

The harmonic series and the divergence test

The harmonic series is

${\sum }_{n=1}^{\infty }\frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots$

This diverges to infinity, even though the terms get arbitrarily small. The proof: group terms as $1,1/2,(1/3+1/4),(1/5+\dots +1/8),\dots$ Each group is at least $1/2$, and there are infinitely many groups.

This is one of the most counterintuitive results in calculus. Terms going to zero is necessary but not sufficient for convergence.

Divergence test (the contrapositive): If ${\lim }_{n\to \infty }{a}_n\ne 0$, then $\sum a_n$ diverges.

This is useful only to rule out convergence. If $\lim a_n=0$, you cannot conclude anything — you need other tests.

The convergence tests

A small set of tests handles almost any series you encounter.

The ratio test

If ${\lim }_{n\to \infty }\left|\frac{a_{n+1}}{a_n}\right|=L$:

• $L<1$: series converges.
• $L>1$: series diverges.
• $L=1$: test is inconclusive.

Example 2: Test ${\sum }_{n=1}^{\infty }\frac{n!}{n^n}$.

Compute the ratio:

$\frac{a_{n+1}}{a_n}=\frac{(n+1)!/(n+1{)}^{n+1}}{n!/n^n}=\frac{(n+1)!}{n!}\cdot \frac{n^n}{(n+1{)}^{n+1}}=\frac{n+1}{n+1}\cdot \frac{n^n}{(n+1{)}^n}={\left(\frac{n}{n+1}\right)}^n$

As $n\to \infty$, this approaches $1/e\approx 0.368$. Since $1/e<1$, the series converges.

The root test

If ${\lim }_{n\to \infty }\sqrt[n]{|a_n|}=L$:

• $L<1$: converges.
• $L>1$: diverges.
• $L=1$: inconclusive.

Useful when $a_n$ contains $n$th powers.

Example 3: Test ${\sum }_{n=1}^{\infty }\frac{1}{(\ln n{)}^n}$.

$\sqrt[n]{|a_n|}=\frac{1}{\ln n}\to 0$. Since $0<1$, the series converges.

The comparison test

If $0\le a_n\le b_n$ for all $n$:

• If $\sum b_n$ converges, so does $\sum a_n$.
• If $\sum a_n$ diverges, so does $\sum b_n$.

In words: comparing to a known series. The harder version is the limit comparison test: if ${\lim }_{n\to \infty }{a}_n/b_n=c>0$ (finite), then $\sum a_n$ and $\sum b_n$ either both converge or both diverge.

Example 4: Test ${\sum }_{n=1}^{\infty }\frac{1}{n^2+n}$.

Compare to $\sum 1/n^2$. The ratio $\frac{1/(n^2+n)}{1/n^2}=\frac{n^2}{n^2+n}\to 1$. So both series converge (since $\sum 1/n^2$ converges to ${\pi }^2/6$).

The integral test

If $f(x)$ is positive, decreasing, and continuous for $x\ge 1$, and $a_n=f(n)$:

${\sum }_{n=1}^{\infty }{a}_n\text{ converges}⟺{\int }_1^{\infty }f(x)dx\text{ converges}$

Example 5 (the p-series): Test ${\sum }_{n=1}^{\infty }\frac{1}{n^p}$ for various $p$.

By integral test, compare to ${\int }_1^{\infty }\frac{1}{x^p}dx$. This converges if $p>1$ and diverges if $p\le 1$. So the p-series converges for $p>1$ and diverges for $p\le 1$.

This is one of the most-used convergence facts in practice.

The alternating series test

For an alternating series ${\sum }_{n=1}^{\infty }(-1{)}^n{b}_n$ (where $b_n>0$): if $b_n$ is decreasing and $\lim b_n=0$, the series converges.

Example 6: ${\sum }_{n=1}^{\infty }\frac{(-1{)}^{n+1}}{n}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots =\ln 2$.

The series converges (alternating series test) even though the absolute series $\sum 1/n$ diverges. This is called conditional convergence.

Absolute vs conditional convergence

A series $\sum a_n$ converges absolutely if $\sum |a_n|$ converges. It converges conditionally if $\sum a_n$ converges but $\sum |a_n|$ diverges.

Absolute convergence is stronger. Absolutely convergent series can be rearranged without changing the sum. Conditionally convergent series can be rearranged to converge to any real number (Riemann rearrangement theorem) — a striking and counterintuitive result.

Power series

A power series is a series of the form

${\sum }_{n=0}^{\infty }{c}_n(x-a{)}^n$

where $c_n$ are coefficients and $a$ is a center point. The radius of convergence $R$ is the value such that the series converges for $|x-a|<R$ and diverges for $|x-a|>R$.

The radius can be computed by the ratio test: $R={\lim }_{n\to \infty }\left|\frac{c_n}{c_{n+1}}\right|$.

Taylor series

Any "nice enough" function $f$ can be written as a Taylor series around a point $a$:

$f(x)={\sum }_{n=0}^{\infty }\frac{f^{(n)}(a)}{n!}(x-a{)}^n=f(a)+f^{\prime }(a)(x-a)+\frac{f^{\prime \prime }(a)}{2!}(x-a{)}^2+\cdots$

This is one of the most useful tools in applied mathematics. It approximates complicated functions by polynomials.

Common Taylor series (around $a=0$):

$e^x={\sum }_{n=0}^{\infty }\frac{x^n}{n!}=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots$

$\sin x={\sum }_{n=0}^{\infty }\frac{(-1{)}^n{x}^{2n+1}}{(2n+1)!}=x-\frac{x^3}{6}+\frac{x^5}{120}-\cdots$

$\cos x={\sum }_{n=0}^{\infty }\frac{(-1{)}^n{x}^{2n}}{(2n)!}=1-\frac{x^2}{2}+\frac{x^4}{24}-\cdots$

$\frac{1}{1-x}={\sum }_{n=0}^{\infty }{x}^n=1+x+x^2+\cdots (|x|<1)$

$\ln (1+x)={\sum }_{n=1}^{\infty }\frac{(-1{)}^{n+1}{x}^n}{n}=x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots (|x|\le 1,x\ne -1)$

Calculators compute $\sin$, $\cos$, $e^x$, and other transcendental functions by evaluating Taylor series to sufficient precision.

Common pitfalls

Pitfall 1 — Confusing the divergence test with a sufficient condition. If $\lim a_n=0$, you cannot conclude convergence. The harmonic series is the famous counterexample.

Pitfall 2 — Misapplying the ratio test. It's inconclusive when $L=1$. Don't conclude anything in that case — try another test.

Pitfall 3 — Rearranging conditionally convergent series. The order matters. If you rearrange a conditionally convergent series, you can change its sum. Be careful in computations.

Pitfall 4 — Forgetting the radius of convergence for Taylor series. $\ln (1+x)$'s Taylor series only works for $|x|\le 1$ (and $x\ne -1$). Using it for $x=2$ gives nonsense.