series: estimate sum within .01

[rcmango]

1. The problem statement, all variables and given/known data

How many terms of the series
infinity
E n =1

1/(1+n^2) must be added to estimate the sum within 0.01?

2. Relevant equations


3. The attempt at a solution

need help please. Also, the answer i believe it 100 terms. However i need to show work to support this answer.

[StatusX]

If we denote the infinite sum by S, ie:

S=\sum_{n=1}^\infty \frac{1}{n^2+1}

and the partial sum of the first N terms by SN:

S_N=\sum_{n=1}^N \frac{1}{n^2+1}

Then the error induced by estimating the infinite sum by the partial sum of the first N terms is:

S-S_N=\sum_{n=N+1}^\infty \frac{1}{n^2+1}

Can you find an upper bound for this sum? Here's a hint: 1/(n-1)-1/n=1/n(n-1).

[rcmango]

while plugging numbers into n in the equation, i can see that the equation appears to approach to 0.

whats next :)

[HallsofIvy]

Yes, of course it approaches 0! "Next" is to answer your question: how large does n have to be to make it less than 0.01?