Sequences, series, summability Proposition

Sum of a finite geometric series

Statement

Let $r$ be a real number with $r \ne 1$ , and let $n$ be a natural number. Then $\sum_{k=0}^{n-1} r^k = 1 + r + r^2 + \cdots + r^{n-1} = \frac{1 - r^n}{1 - r}.$

Remarks

The proof multiplies the sum $S$ by $1 - r$ . The product $(1 - r) S$ telescopes to $1 - r^n$ . For $r = 1$ the sum is $n$ . The identity holds in each commutative ring in the form $(1 - r) \sum_{k=0}^{n-1} r^k = 1 - r^n$ , with no division. Mathlib also states the version for a sum over the indices $m \le k < n$ , as geom_sum_Ico. Letting $n \to \infty$ gives the geometric series.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Sequence definition
Real sequence definition
Geometric sequence definition
Sum of a finite geometric series statement

Where this leads

109 pages build on this one.

Tag	`05APWSX53S`
URI	`m2://05APWSX53S`
Kind	statement, proposition
Writing	W0 stub
MSC	40A05
Status	This copy of the site has no build data. Run `m2 status 05APWSX53S`.
Uses	Geometric sequence

Sum of a finite geometric series

Statement

Remarks

Reading path

Where this leads

Further reading

What this page needs for W1

What links here

Used by