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.

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- Jan 30th 2007, 11:52 PMrcmangoseries: estimate sum within .01
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. - Jan 31st 2007, 04:08 AMtopsquark
- Jan 31st 2007, 04:45 AMCaptainBlack
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