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March 29, 2017

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The sum from 0 to infinity of (-1)^n(3/3^n) is convergent or divergent? If convergent, what is the sum?

I got that it's convergent and the sum is 9/2, but that's wrong.

  • calculus - ,

    It is not clear without sufficient parentheses what the expression really is.
    I assume it to be:
    Sum((-1)^n * (3/3^n)) for n=0 -> ∞
    This is an alternating geometric series.

    (9/2) is the correct sum for the geometric series (non-alternating).

    Write out the first few terms of the series:
    3 - 3/3 + 3/9 - 3/27 + 3/81 - ...
    which can be regrouped into two geometric series:
    3(1+1/9+1/81+...) - (1+1/9+1/81+...)
    =3(9/8) - (9/8)
    =2(9/8)
    =9/4

    Note that if the minus sign becomes a plus sign, we get the geometric sum of 9/2.

  • calculus - ,

    Oh, right . I forgot all about the -1. Thanks.

  • calculus - ,

    You're welcome!

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