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Math Help - Find all real numbers such that the series converges!

  1. #1
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    Find all real numbers such that the series converges!

    Is there a real number c such that the series:

    ∑ (e - (1+ 1/n)^n + c/n), where the series goes from n=1 to n=∞, is convergent?

    I tried to use ratio test but it turned out really messy. Any other ideas?

    Thanks.
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  2. #2
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    Re: Find all real numbers such that the series converges!

    deleted
    Last edited by romsek; February 20th 2014 at 11:21 PM. Reason: probably wrong
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  3. #3
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    Re: Find all real numbers such that the series converges!

    Yes I knew that and I did come to the harmonic series conclusion, but isn't it true that lim as n→∞ (1+1/n)^n is different from summation (1+1/n)^n as n from 1 to infinity?
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  4. #4
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    Re: Find all real numbers such that the series converges!

    yeah ok, let me think about it some.
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    Re: Find all real numbers such that the series converges!

    Sure thanks!
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  6. #6
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    Re: Find all real numbers such that the series converges!

    Also since you are answering all my questions, I might as well post one more for you to look at. That doesn't look too hard, but there's got to be some trick that I can't think of at the moment!
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  7. #7
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    Re: Find all real numbers such that the series converges!

    Quote Originally Posted by mynameisanthonyg2013 View Post
    Is there a real number c such that the series:
    ∑ (e - (1+ 1/n)^n + c/n), where the series goes from n=1 to n=∞, is convergent?
    I tried to use ratio test but it turned out really messy. Any other ideas?
    In the case that $c>0 $ then:
    It is well known that ${\left( {1 + {n^{ - 1}}} \right)^n}$ is a monotone increasing sequence $\to e$.
    Therefore $(\forall n)\left[e-{\left( {1 + {n^{ - 1}}} \right)^n}>0\right]$. So use the comparison test with $\sum\dfrac{c}{n}$.
    Last edited by Plato; February 21st 2014 at 10:45 AM.
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  8. #8
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    Re: Find all real numbers such that the series converges!

    So use the compassion test with \sum \frac{c}{n}.
    do you mean comparison test?
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