## Sum

The sum of a geometric series is finite as long as the terms approach zero; as the numbers near zero, they become insignificantly small, allowing a sum to be calculated despite the series being infinite. The sum can be computed using the self-similarity of the series.### Example

*s*from the original series

*s*cancels every term in the original but the first:

### Formula

For , the sum of the first*n*terms of a geometric series is:

*a*is the first term of the series, and

*r*is the common ratio. We can derive this formula as follows:

*n*goes to infinity, the absolute value of

*r*must be less than one for the series to converge. The sum then becomes

*a*= 1, this simplifies to:

*r*. We can derive this formula:

*a*.

This formula is only valid for convergent series (i.e., when the magnitude of

*r*is less than one). For example, the sum is undefined when

*r*= 10, even though the formula gives

*s*= −1/9.

This reasoning is also valid, with the same restrictions, for the complex case.

### Proof of convergence

We can prove that the geometric series converges using the sum formula for a geometric progression:*r*+

*r*

^{2}+ ... +

*r*

^{n})(1−

*r*) = 1−

*r*

^{n+1}and

*r*

^{n+1}→ 0 for |

*r*| < 1.

Convergence of geometric series can also be demonstrated by rewriting the series as an equivalent telescoping series. Consider the function g(K) = (r^(K+1))/(1-r). Note that: r = g(0) - g(1), r^2 = g(1) - g(2), r^3 = g(2) - g(3), . . . Thus: S = r + r^2 + r^3 + . . . = (g(0) - g(1)) + (g(1) - g(2)) + (g(2) - g(3)) + . . . If |r|<1, then g(K) -> 0 as K -> infinity, and so S converges to g(0) = r/(1-r).

Applications

### Repeating decimals

Main article: Repeating decimal

A repeating decimal can be thought of as a geometric series whose common ratio is a power of 1/10. For example:^{n}- 1.

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