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Series: estimate sum within .01

  1. Jan 30, 2007 #1
    1. The problem statement, all variables and given/known data

    How many terms of the series
    infinity
    E n =1

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

    2. Relevant equations


    3. 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.
     
  2. jcsd
  3. Jan 30, 2007 #2

    StatusX

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    Homework Helper

    If we denote the infinite sum by S, ie:

    [tex]S=\sum_{n=1}^\infty \frac{1}{n^2+1} [/tex]

    and the partial sum of the first N terms by SN:

    [tex]S_N=\sum_{n=1}^N \frac{1}{n^2+1} [/tex]

    Then the error induced by estimating the infinite sum by the partial sum of the first N terms is:

    [tex]S-S_N=\sum_{n=N+1}^\infty \frac{1}{n^2+1} [/tex]

    Can you find an upper bound for this sum? Here's a hint: 1/(n-1)-1/n=1/n(n-1).
     
  4. Jan 30, 2007 #3
    while plugging numbers into n in the equation, i can see that the equation appears to approach to 0.

    whats next :)
     
  5. Jan 31, 2007 #4

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    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?
     
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