Series Sums via Partial Sums

Summary

The sum of an infinite series is defined as the limit of its partial sums. Closed forms for $S_{n}$ (geometric, telescoping) make convergence transparent.

Prerequisites

Sequences

Definition

For $\sum_{k = 1}^{\infty} a_{k}$ , the $n$ th partial sum is

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

The series converges to $S$ if and only if $lim_{n \to \infty} S_{n} = S$ .

Worked Example

Geometric series with $a = 1$ , $r = 1 / 2$ :

S_{n} = \sum_{k = 0}^{n - 1} (\frac{1}{2})^{k} = \frac{1 - (1 / 2)^{n}}{1 - 1 / 2} = 2 (1 - 2^{- n}) \to 2.

General geometric partial sum ( $r \neq 1$ ):

S_{n} = a \frac{1 - r^{n}}{1 - r} .

Common Mistakes

Identifying the series sum with a single term $a_{n}$ .
Using $S_{n} = a / (1 - r)$ for finite $n$ without the $r^{n}$ correction.

Connections

Geometric Series, Telescoping Series, Infinite Series

References

Partial sums define series convergence in OpenStax Calculus Volume 2.^[1]

OpenStax, Calculus Volume 2, Section 5.2, https://openstax.org/details/books/calculus-volume-2 ↩︎