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Series approximation

  1. Apr 4, 2009 #1
    1. The problem statement, all variables and given/known data

    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.

    2. Relevant equations

    I think Rn = s - sn

    3. 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 ?
     
    Last edited: Apr 4, 2009
  2. jcsd
  3. Apr 4, 2009 #2

    Tom Mattson

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    Check your book for a remainder theorem associated with the integral test. Is it there?
     
  4. Apr 4, 2009 #3
    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 ?
     
  5. Apr 4, 2009 #4

    Tom Mattson

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

    [tex]R_N\leq a_N+\int_N^{\infty}f(x)dx[/tex]

    And yes, it will help for part c.
     
  6. Apr 4, 2009 #5
    Uha, thanks alot..

    But, I still wondering when do we use two boundries and when to use only one when computing the error ??
     
  7. Apr 4, 2009 #6

    Tom Mattson

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    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 [itex]R_N[/itex].
     
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