TITLE: What's the sum of this series?
QUESTION [7 upvotes]: I would like to know how to find out the sum of this series:
$$1 - \frac{1}{2^2} + \frac{1}{3^2} - \frac{1}{4^2} + \frac{1}{5^2} - \frac{1}{6^2} + \cdots$$
The answer is that it converges to a sum between $\frac 34$ and $1$, but how should we go about estimating this sum? 
Thanks!

REPLY [3 votes]: If we did not know an exact answer, here is a crude way to estimate the sum of the series.  
Our series is an alternating series, with terms that decrease in absolute value and have limit $0$. So the error made by truncating at a particular term has absolute value less than the absolute value of the first "neglected" term. 
For example, if we use $1-\frac{1}{2^2}+\frac{1}{3^2}-\frac{1}{4^2}+\frac{1}{5^2}$ as an approximation, then the absolute value of the error is less than $\frac{1}{6^2}$. Furthermore, the error is "negative," that is, our estimate is greater than the true value.