Series: estimate sum within .01

[rcmango]

Homework Statement

How many terms of the series
infinity
E n =1

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

Homework Equations

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 S_N:

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

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?