SumThe 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.
FormulaFor , the sum of the first n terms of a geometric series is:
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 convergenceWe can prove that the geometric series converges using the sum formula for a geometric progression:
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).
Repeating decimalsA repeating decimal can be thought of as a geometric series whose common ratio is a power of 1/10. For example:
To Join Ajit Mishra's Online Classroom CLICK HERE