How many terms of the series
infinity
E
n =1
1/(1+n^2) must be added to estimate the sum within 0.01?
struggling with this one.
As the summand is a monotonic decreasing positive function of n, we can bound the
remainder as:
$\displaystyle
\sum_1^{\infty}\frac{1}{(1+n^2)}= \sum_1^{N}\frac{1}{1+n^2}+\sum_{N+1}^{\infty}\frac {1}{1+n^2}
$$\displaystyle
\le \sum_1^{N}\frac{1}{1+n^2}+\int_N^{\infty} \frac{1}{1+n^2} dn
$
So if we choose $\displaystyle N$ such that
$\displaystyle \int_N^{\infty} \frac{1}{1+n^2} dn \le 0.01$
Then the error in the truncated series will be $\displaystyle \le 0.01$ as required.
RonL