How Do I Calculate the Partial Sum of an Alternating Series?

[Nick_L]

Can anyone help me out with calculating the partial sum of an alternating series? For example, how would I find the sum correct to 4 decimal places of:

What I tried was finding how many terms it would take the have an error that was < .0001 then found the sum with that many terms... I got 0.10969 as the partial sum using 4 terms.

 

[yyat]

Note that

$$\sum_{n=1}^{\infty}nx^n=\frac{x}{(x-1)^2}$$

which can be obtained from the geometric series by computing the derivative and multiplying by x. Hence

$$\sum_{n=1}^{\infty}n(-\frac{1}{11})^n=-\frac{11}{144}$$