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Math Help - Partial Sum Approximation for Alternating Harmonic Series Question

  1. #1
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    Partial Sum Approximation for Alternating Harmonic Series Question

    1. The problem statement, all variables and given/known data

    Find a value for n for which the nth partial sum is ensured to approximate the sum of the alternating harmonic infinite series to three decimal places.

    2. Relevant equations

    Sn = Ʃ(-1)^k+1*1/k = 1 - 1/2 + 1/3 - 1/4 + 1/5 - . . .

    S1 = 1
    S2 = 1 - 1/2
    S3 = 1 - 1/2 + 1/3
    S4 = 1 - 1/2 + 1/3 -1/4

    .
    .
    .

    Sn = ?

    3. The attempt at a solution

    An attempt was made to derive a formula that would permit finding the value of the nth partial sum (e.g., S1000 = ?), but without success. It appears as though such a formula might not exist. Any help in this regard would be most appreciated.
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  2. #2
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    Re: Partial Sum Approximation for Alternating Harmonic Series Question

    You should be aware of a theorem about alternating series - that when |a_n| is nonincreasing, \left| \sum_{n=1}^\infty a_n - \sum_{n=1}^N a_n \right| \le |a_{N+1}|. So you just need to find an N for which |a_{N+1}| is small enough that you can determine \sum_{n=1}^\infty a_n to within 3 decimal places.

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