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Math Help - Leibniz Test

  1. #1
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    Leibniz Test

    Infinity over Sigma n=1 ((-1)^n)*(n)*(e^-n) Now I know Im suppose to use the leibniz test and I got as far as lanl=n/e^n but Im not sure where to go from there.
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  2. #2
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    Well the limit is zero, so you need to prove that is a strictly decreasing sequence.

    Put f(x)=\frac x{e^x}, so f'(x)=\frac{{{e}^{x}}-x{{e}^{x}}}{{{e}^{2x}}}=\frac{{{e}^{x}}(1-x)}{{{e}^{2x}}}<0, for x\ge2.

    By the alternating series test, your series converges.
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  3. #3
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    Could I plug n+1 for all n as well to prove that its decreasing?

    n+1/e^(n+1) < n/e^(n)
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  4. #4
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    Yes but you need to prove that such inequality holds.

    The best and fast way to prove that general term is a strictly decreasing sequence is to contemplate its derivative as I did above.
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