Sum of Series (1/n^2) Approximation & Error Estimation

[remaan]

Homework Statement

Using the sum of the first 10 terms ,
Estimate the sum of the series (1/n^2) n from 1 to infinity ? How good the estimate is ?

c) Find a value for n that will ensure that the error in the approximation s= sn is less than .001.

Homework Equations

I think Rn = s - sn

The Attempt at a Solution

Am ok with that, but how to know who good the appr. is ??
And do we do the integration to find the value of n ? if so please tell how ?

 

[quantumdude]

Check your book for a remainder theorem associated with the integral test. Is it there?

 

[remaan]

Uha, I think that you mean this formula

Sn+ ∫_(n+1 )^(infity )▒〖f(x)dx<s<Sn+ ∫_n^(infinity )▒f(x)dx〗

Do you think that this works for part c ?

 

[quantumdude]

Your equation doesn't appear as it should on my browser, but there's a simple theorem for estimating the remainder $R_N$ that exists when the nth partial sum $S_N$ of a convergent series is computed. It says:

$$R_N\leq a_N+\int_N^{\infty}f(x)dx$$

And yes, it will help for part c.

 

[remaan]

Uha, thanks a lot..

But, I still wondering when do we use two boundries and when to use only one when computing the error ??

 

[quantumdude]

You're not computing the error, you're estimating it. And any series that converges by the integral test has only positive terms. So there is always an implicit lower boundary of 0 on $R_N$.