Series

Overview

Let $(a_{n})_{n \geq 0}$ be a sequence. Then the sequence of partial sums of $(a_{n})_{n \geq 0}$ refers to a new sequence $(s_{n})_{n \geq 0}$ given by

s_{n} = a_{0} + a_{1} + \dots + a_{n} = \sum_{k = 0}^{n} a_{k} .

A sequence of partial sums is called an (infinite) series. Such a series is often represented as $\sum_{k = 0}^{\infty} a_{k} .$

A tail of $\sum a_{k}$ refers to the same sum with a finite number of initial terms omitted.

Convergence

If there is a real (or complex) number $S$ such that

lim_{n \to + \infty} s_{n} = S,

we say that series $\sum_{k = 1}^{\infty} a_{k}$ is convergent and has the sum $S$ . If $(s_{n})$ diverges, we say the series diverges and has no sum.

Linearity

Let $α, β \in C$ and both $\sum a_{n}$ and $\sum b_{n}$ be convergent infinite series of complex terms. Then the series $\sum (α a_{n} + β b_{n})$ also converges and its sum is given by equation

\sum_{k = 1}^{\infty} (α a_{n} + β b_{n}) = α \sum_{k = 1}^{\infty} a_{n} + β \sum_{k = 1}^{\infty} b_{n} .

Monotone Convergence

Let $n_{0} \in N$ and assume that $a_{n} \geq 0$ for each $n \geq n_{0}$ . Then the series $\sum a_{n}$ converges if and only if the sequence of its partial sums is bounded above.

Absolute Convergence

A series $\sum a_{n}$ is called absolutely convergent if $\sum | a_{n} |$ converges. The series is conditionally convergent if $\sum a_{n}$ converges but $\sum | a_{n} |$ diverges.

Assume $\sum | a_{n} |$ converges. Then $\sum a_{n}$ also converges, and we have

| \sum_{n = 1}^{\infty} a_{n} | \leq \sum_{n = 1}^{\infty} | a_{n} | .

Term Test

If the series $\sum a_{n}$ converges, then its $n$ th term tends to $0$ . That is,

lim_{n \to + \infty} a_{n} = 0.

Direct Comparison Test

Let $n_{0} \in N$ and $(a_{n})$ and $(b_{n})$ be sequences such that $a_{n}, b_{n} \geq 0$ for all $n \geq n_{0}$ . If there exists a positive constant $c$ such that $a_{n} \leq c b_{n}$ for all $n \geq n_{0}$ , then convergence of $\sum b_{n}$ implies convergence of $\sum a_{n}$ .

In such a situation, we say that series $\sum b_{n}$ dominates series $\sum a_{n}$ .

Limit Comparison Test

Let $n_{0} \in N$ and assume $a_{n} \geq 0$ , $b_{n} > 0$ for all $n \geq n_{0}$ . Also suppose that

lim_{n \to + \infty} \frac{a_{n}}{b_{n}} = c, c > 0.

Then $\sum a_{n}$ converges if and only if $\sum b_{n}$ converges.

Integral Test

Let $a \in N$ and $f$ be a positive decreasing function, defined for all real $x \geq a$ . For each $n \geq a$ , let

s_{n} = \sum_{k = a}^{n} f (k) and t_{n} = \int_{a}^{n} f (x) d x .

Then both sequences $(s_{n})$ and $(t_{n})$ converge or both diverge.

P-Test

Let $a \in Z^{+}$ and $b \in R^{+}$ . A $p$ -series and $p$ -integral are those series and integrals of the following respective forms:

\sum_{n = a}^{\infty} \frac{1}{n^{p}} and \int_{b}^{\infty} \frac{1}{x^{p}} d x .

The $p$ -test for series (or $p$ -series test) states that such a series converges if and only if $p > 1$ . Likewise, the $p$ -test for integrals (or $p$ -integral test) states that such an integral converges if and only if $p > 1$ .

Root Test

Let $\sum a_{n}$ be a series of nonnegative terms such that

a_{n}^{1 / n} \to R as n \to + \infty .

If $R < 1$ , the series converges.
If $R > 1$ , the series diverges.
If $R = 1$ , the test is inconclusive.

Ratio Test

Let $\sum a_{n}$ be a series of positive terms such that

\frac{a_{n + 1}}{a_{n}} \to L as n \to + \infty .

If $L < 1$ , the series converges.
If $L > 1$ , the series diverges.
If $L = 1$ , the test is inconclusive.

Dirichlet's Test

Let $\sum a_{n}$ be a series of complex terms whose partial sums form a bounded sequence. Let $(b_{n})$ be a decreasing sequence which converges to $0$ . Then the series $\sum a_{n} b_{n}$ converges.

Alternating Series Test

Let $\sum_{n = 1}^{\infty} (- 1)^{n - 1} a_{n}$ be an alternating series. The alternating series test states that such a series converges if

$| a_{n} |$ decreases monotonically, and
$a_{n} \to 0$ as $n \to + \infty$ .

Let $S$ denote the sum of such a series and $s_{n}$ denote the $n$ th partial sum. Then

0 \leq | s_{n} - S | \leq | a_{n + 1} | .

Abel's Test

Let $\sum a_{n}$ be a convergent series of complex terms and let $(b_{n})$ be a monotonic convergent sequence of terms. Then the series $\sum a_{n} b_{n}$ converges.

Sum Functions

Let $(f_{n})$ be a sequence of functions for which each $f_{n} (x)$ is a partial sum of another function, say

f_{n} (x) = \sum_{k = 1}^{n} u_{k} (x) .

Let $S$ denote the set of points $x$ for which sequence $(f_{n} (x))$ converges. The function

f (x) = lim_{n \to \infty} f_{n} (x) = \sum_{k = 1}^{\infty} u_{k} (x) if x \in S,

is called the sum function of the sequence $(f_{n})$ . We say the series $\sum u_{k}$ converges pointwise to $f$ on $S$ .

Weierstrass M-Test

Let $\sum u_{n}$ be a series of functions which converges pointwise to a function $f$ on a set $S$ . If there is a convergent series of positive constants $\sum M_{n}$ such that

0 \leq | u_{n} (x) | \leq M_{n} for every n \geq 1 and every x \in S,

then the series $\sum u_{n}$ converges uniformly on $S$ .

Special Cases

Alternating

An alternating series is one in which each consecutive term alternates in sign. That is, given $a_{n} > 0$ for all $n \geq 1$ , series of form

\sum_{n = 1}^{\infty} (- 1)^{n - 1} a_{n} = a_{1} - a_{2} + a_{3} - a_{4} + \dots + (- 1)^{n - 1} a_{n} + \dots

Arithmetic

Let $(a_{n})_{n \geq 0}$ be an arithmetic sequence. Then the partial sums are given by

\sum_{k = 0}^{n} a_{k} = \frac{(a_{0} + a_{n}) (n + 1)}{2} .

Unless the common difference is $0$ , such a series always diverges.

Geometric

Let $(a_{n})_{n \geq 0}^{r}$ be a geometric sequence. Then the partial sums are given by

\sum_{k = 0}^{n} a_{k} = \frac{a_{0} (1 - r^{n + 1})}{1 - r} .

A geometric series converges if and only if its common ratio $r$ satisfies $| r | < 1$ . If $| r | < 1$ , then

\sum_{k = 0}^{\infty} a_{k} = \frac{a_{0}}{1 - r} .

Harmonic

The harmonic series refers to the divergent series

\sum_{k = 1}^{\infty} \frac{1}{k} .

Telescoping

Telescoping refers to the property of summations in which consecutive terms cancel out. The telescoping property states that

\sum_{k = 1}^{n} (a_{k} - a_{k + 1}) = a_{1} - a_{n + 1} .

Let $(a_{n})_{n \geq 0}$ and $(b_{n})_{n \geq 0}$ be two sequences of complex numbers such that

a_{n} = b_{n} - b_{n + 1} for n = 1, 2, 3, \dots

Then the series $\sum a_{n}$ converges if and only if the sequence $(b_{n})$ converges, in which case we have

\sum_{n = 1}^{\infty} a_{n} = b_{1} - L, where L = lim_{n \to + \infty} b_{n} .