Consider the series given by : $$\sum _{ n=1 }^{ \infty }{ { e }^{ -n } } $$

It can be shown that this series converges using the integral test. While the integral test can be used to show convergence, the value of the integral does not tell us what value the series converges to.

Given that the series converges, say to $S$, we can define the "remainder" as $R=S-{ S }_{ n }$, where $S$ is the sum of the series and ${S}_{n}$ is the ${ n }^{ th }$ partial sum for the series. It can be shown,

$$R\le \int _{ N }^{ \infty }{ f(x)dx }$$

where $N$ represents the number of terms added in order to obtain an approximation for $S$. Use this fact to find the least number of terms of the given series needed to obtain a sum that is within 0.00001 of the limit for the series.