series: estimate sum within .01

**rcmango** · January 30th 2007, 11:52 PM

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.

**topsquark** · January 31st 2007, 04:08 AM

Originally Posted by rcmango

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.

If you are going post these may I suggest LaTeX?
$\sum_{n = 1}^{\infty} \frac{1}{1 + n^2}$

Click on the "quote" button to see how I coded this.

-Dan

**CaptainBlack** · January 31st 2007, 04:45 AM

Originally Posted by rcmango

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:

$<br /> \sum_1^{\infty}\frac{1}{(1+n^2)}= \sum_1^{N}\frac{1}{1+n^2}+\sum_{N+1}^{\infty}\frac {1}{1+n^2} <br />$ $<br /> \le \sum_1^{N}\frac{1}{1+n^2}+\int_N^{\infty} \frac{1}{1+n^2} dn<br />$

So if we choose $N$ such that

$\int_N^{\infty} \frac{1}{1+n^2} dn \le 0.01$

Then the error in the truncated series will be $\le 0.01$ as required.

RonL

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