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Math Help - infinite series--integral?

  1. #1
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    infinite series--integral?

    Hi, I am supposed to determine whether "the sum of the sequence a(n)from n=1 to n=infinity" converges (i.e. whether or not the sequence is summable) where a(n)=1/[n^(1+(1/n)]. I figure I can use the fact that this sequence is summable if lim(A->infinity)[integral from 1 to A of f(x)] exists where f(x)=1/[x^(1+(1/x))]. My problem is I forget (or maybe never knew) how to find this integral which I must do to find the limit of the integral as A approaches infinity. Am I even approaching the question in the correct way?

    Thankyou for any help
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  2. #2
    MHF Contributor chisigma's Avatar
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    Once You have the explicit expressione for the a_{n}, finding the integral for the covergence test is very easy: You have only to swap the n and the x...

    In Your example...

    a_{n} = \frac{1}{n^{1+\frac{1}{n}}} \rightarrow f(x)= \frac{1}{x^{1+\frac{1}{x}}} (1)

    Kind regards

    \chi \sigma
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  3. #3
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    Quote Originally Posted by chisigma View Post
    Once You have the explicit expressione for the a_{n}, finding the integral for the covergence test is very easy: You have only to swap the n and the x...

    In Your example...

    a_{n} = \frac{1}{n^{1+\frac{1}{n}}} \rightarrow f(x)= \frac{1}{x^{1+\frac{1}{x}}} (1)

    Kind regards

    \chi \sigma

    Thankyou for assuring me that I am going in the right direction with the integral test. I know I have to swap the n and the x and find the integral of f(x)= \frac{1}{x^{1+\frac{1}{x}}} (1) but my question is how do I do that? I can't use integration by parts because logx is missing from the equation (the derivative of 1+ 1/x would be logx right?) so I don't know how to integrate this.
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