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Math Help - series: estimate sum within .01

  1. #1
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    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.
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by rcmango View Post
    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
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by rcmango View Post
    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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