Finding Error in Alternating Series

[SclayP]

The problem statement says to find out if the next series converge, and if it does to calculate the sum with an error $$ε< 10^{-3}$$

The serie is this one

$\sum^{\infty}_{n=1} (-1)^nne^{-n}$
First of all the serie converges because of Leibniz Criterion but the i did the series of |a_n|

I did it with Cauchy Criterion and the seris converges again...

$\sum_{n=1}^{\infty} \frac{n}{e^{-n}}$

$$\lim_{n \rightarrow +\infty} \frac{\sqrt[n]{n}}{\sqrt[n]{e^n}}$$

$$\lim_{n \rightarrow +\infty} \frac{\sqrt[n]{n}}{e}$$

$$\frac{1}{e}<1$$

Now i have to find the error and that i don't know how to do it..

Thank.

 

[Dick]

With an alternating series whose terms are decreasing there's an easy method. If you sum the first m terms of your series then the error is less the absolute value of the m+1 term.