real analysis question. there are definition and example may help.

Example 1 A series of the form En ar" for constants a and r is called a geometric series. These are the easiest series to sum. For r # 1, the partial sums $, are given by n Dark = I- potl 1 - r (1) k=0 This identity can be verified by mathematical induction or by mul- tiplying both sides by 1-r, in which case the right hand side equals a - art and the left side becomes n n (1-r) Cark = Lark - Lark+ k=0 k=0 K=0 = atartars+ .. . + arm - (ar tar= + .. . + ar" + arntly = a - arn+1 For r 0 there exists a number NV such that m, n > NV implies S, - 8m SE.Suppose (Sn )ney is a sequence such that for some fixed A > 0 and r E (0, 1) the sequence satisfies |Sn+1 - Sn n for convenience in the definition of a Cauchy sequence without changing the meaning of the definition.)