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Math Help - approximate the sum

  1. #1
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    approximate the sum

    Show that if we want to approximate the sum of the series N_infinity n^-1.001 so that the error is less than 5 in the ninth decimal place, then we need to add more than 10^11.301 terms.

    need help please.
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  2. #2
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    Quote Originally Posted by rcmango View Post
    Show that if we want to approximate the sum of the series N_infinity n^-1.001 so that the error is less than 5 in the ninth decimal place, then we need to add more than 10^11.301 terms.

    need help please.
    sum_{n=1 to infty} 1/n^1.001 = sum_{n=1 to N} 1/n^1.001 + sum_{n=N+1 to infty) 1/n^1.001,

    but:

    int_{x=N1 to infty} 1/x^1.001 dx < sum_{n=N+1 to infty) 1/n^1.001 < int_{x=N to infty} 1/x^1.001 dx

    so if we choose N so that int_{x=N to infty} 1/x^1.001 dx <= 0.000000005
    then we are done. The integral is 1000/N^0.001, so we seek N such that:

    5 10^-9 = 1000/N^0.001

    which has solution:

    N ~= 1.07 10^11301

    which is more than 10^11301 terms.

    RonL
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