# series: estimate sum within .01

• Jan 30th 2007, 11:52 PM
rcmango
series: 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 AM
topsquark
Quote:

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
• Jan 31st 2007, 04:45 AM
CaptainBlack
Quote:

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:

$
\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}
$
$
\le \sum_1^{N}\frac{1}{1+n^2}+\int_N^{\infty} \frac{1}{1+n^2} dn
$

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