5.2 Infinite series  (Page 4/16)

is a geometric series with initial term $a=1$ and ratio $r=1\text{/}2.$

In general, when does a geometric series converge? Consider the geometric series

when $a>0.$ Its sequence of partial sums $\left\{S_k\right\}$ is given by

Consider the case when $r=1.$ In that case,

Since $a>0,$ we know $ak\to \infty$ as $k\to \infty .$ Therefore, the sequence of partial sums is unbounded and thus diverges. Consequently, the infinite series diverges for $r=1.$ For $r\ne 1,$ to find the limit of $\left\{S_k\right\},$ multiply [link] by $1-r.$ Doing so, we see that

All the other terms cancel out.

Therefore,

From our discussion in the previous section, we know that the geometric sequence $r^k\to 0$ if $\left|r\right|<1$ and that $r^k$ diverges if $\left|r\right|>1$ or $r=\text{±}1.$ Therefore, for $\left|r\right|<1,$ $S_k\to a\text{/}(1-r)$ and we have

If $\left|r\right|\ge 1,$ $S_k$ diverges, and therefore

Geometric series sometimes appear in slightly different forms. For example, sometimes the index begins at a value other than $n=1$ or the exponent involves a linear expression for $n$ other than $n-1.$ As long as we can rewrite the series in the form given by [link] , it is a geometric series. For example, consider the series

To see that this is a geometric series, we write out the first several terms:

We see that the initial term is $a=4\text{/}9$ and the ratio is $r=2\text{/}3.$ Therefore, the series can be written as

Since $r=2\text{/}3<1,$ this series converges, and its sum is given by

We now turn our attention to a nice application of geometric series. We show how they can be used to write repeating decimals as fractions of integers.