1.6.4—Geometric convergence

For first term a and common ratio r, S∞=a/(1−r) exists only if |r|<1. The partial sums approach a finite limit because later terms shrink to zero.

Check convergence before using the formula. A negative r gives alternating partial sums, but still converges when |r|<1.

3−1.5+0.75−… has a=3,r=−0.5 and S∞=3/1.5=2; the alternating signs do not prevent convergence.

A ratio close to 1 may converge slowly, while r=1 or −1 does not produce a finite infinite sum.