10.2 Working with Geometric Series

A geometric series is a series in which the ratio between successive terms is constant. If the first term is a and the common ratio is r, then the series has the form ${\sum_{n=0}^{\infty} ar^n}$ A geometric series converges exactly when ${|r|<1}$, and in that case its sum is ${\frac{a}{1-r}}$. If ${|r|\ge1}$, the series diverges because the terms do not shrink in the right way for the partial sums to settle to a finite value. This is one of the most important special series in BC calculus.

You should recognize geometric series quickly by checking whether each term is obtained by multiplying the previous term by the same number. If that ratio exists and stays constant, then the geometric formula applies. This topic is useful because geometric series are among the few infinite series whose convergence and exact sum can often be determined immediately. They also appear later inside power series and Taylor series.