Infinite Series¶

Definition¶

Let $(a_n)_{n=1}^{\infty}$ be a sequence. For any integer $N\geq 1\text{,}$ we define the $N$ -th partial sum, formed by the finite series of the first $N$ terms, $S_N$ of this sequence to be a real number $\displaystyle S_N = \sum_{n=1}^N a_n\text{.}$ 1. If the sequence $(S_N)_{N=1}^\infty$ converges to $L\in\mathbb{R}$ as $N\to\infty\text{,}$ then we say that the infinite series $\displaystyle\sum_{n=1}^\infty a_n$ is convergent and converges to $L\text{.}$ We write $\displaystyle L=\sum_{n=1}^\infty a_n$ and say that $L$ is the sum of the infinite series. 2. If the sequence $(S_N)_{N=1}^\infty$ diverges, then we say $\displaystyle\sum_{n=1}^\infty a_n$ is divergent.

Interpretation¶

An infinite series $\displaystyle \sum_{n=1}^\infty a_n$ adds up infinitely many terms, and the definition gives this a precise meaning: add the first $N$ terms, an ordinary finite sum, and ask whether these partial sums settle down as $N$ grows. The sum of the series is the limit of the sequence $(S_N)$ , not the result of adding infinitely many numbers one by one.

Terms 1/2 to the n shrinking towards 0 and the partial sums rising towards the limit L = 1, drawn as a horizontal line.

The figure shows the geometric series with terms $a_n=\tfrac{1}{2^n}$ $\frac{1}{2 ^{n}}$ . The terms (red squares) shrink towards $0$ , while the partial sums $S_N=1-\tfrac{1}{2^N}$ (blue dots) rise towards the dashed line at $1$ . Each blue dot is the previous one plus the next term, $S_N=S_{N-1}+a_N$ $a_{N}$ , and the series converges to $L=1$ $1$ .

Increase $N$ : a series converges exactly when its partial sums $S_N$ settle. Compare $\frac1n$ , whose terms shrink but whose sums do not settle, with $(-1)^{n+1}$ , whose terms do not shrink at all.

Two sequences are involved and should be kept apart: the terms $(a_n)$ and the partial sums $(S_N)$ . Since $a_N=S_N-S_{N-1}$ $S_{N - 1}$ , a convergent series has $a_N\to L-L=0$ $0$ , which is the necessary condition in the Key Results. The converse fails: the terms of the harmonic series $\displaystyle \sum 1/n$ tend to $0$ , but its partial sums grow past every bound, only slowly.

A common slip is to regroup the terms as if the series were a finite sum. Grouping $(1-1)+(1-1)+\cdots$ suggests the sum $0$ , and $1+(-1+1)+(-1+1)+\cdots$ suggests $1$ . In fact $\displaystyle \sum(-1)^{n+1}$ diverges: its partial sums alternate between $1$ and $0$ .

The same pattern, a limit of finite approximations, returns later in improper integrals.

Key Results¶

A necessary condition for an infinite series $\displaystyle\sum_{n=1}^\infty a_n$ to converge is that its terms $a_n$ tend to zero.
A series may still diverge even if its terms tend to zero.

Examples¶

$\displaystyle \sum_{n=1}^\infty \frac{1}{n}$ diverges.
$\displaystyle \sum_{n=1}^\infty \frac{1}{n^2}$ converges.

Prerequisites¶

Sources¶

Math I: Calculus · Lecture 10

Aliases: infinite sum Symbols: element of, infinity, real numbers, sum

Last updated: 20 October 2026 03:02