Smallest $N$ for which we can guarantee the approximation error of an alternating series
Matthew Barrera 
What is the smallest value $N$ for which we can guarantee that the approximation error of the alternating series
$$S=\sum_{n=1}^\infty\frac{(-1)^n}{n^{7/2}}$$
by the partial sum, $$S_N=\sum_{n=1}^N\frac{(-1)^n}{n^{7/2}}$$ is at most $10^{-2}$?
$\endgroup$ 41 Answer
$\begingroup$This is an alternating series with terms that decrease in absolute value. This means that the sum is always bounded by any two consecutive partial sums and the error is less than the last term used.
So, you know that if $1/n^{7/2} < 10^{-2}$ then this $n$ will be certainly enough. This means $n^{7/2} > 100$ or $n > 100^{2/7} = 10^{4/7} = 3.7... $ so $n \ge 4$ will do.
To find the minimum $n$, get the sums for smaller $n$ and get a more accurate estimate by taking a larger $n$ and see which differs by less than $.01$.
$\endgroup$ 3
