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Math Help - Check series convergence 2

  1. #1
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    Check series convergence 2

    I have SUM [ ((-1)^n * n!)/n^n ] from n=1 to +inf . I need to check both absolute and conditional convergence.

    Now this is very tricky.. the (-1)^n tells me i need to try with the alternating series test.. Than An= n!/n^n , and lim n->inf An is impossible to find for me.

    When i tried with Ratio test for absolute convergence i got 1 as result which tells me nothing .

    My question here is: Can i cancel n! , and say the series will converge is the series SUM [ 1/n^n ] from n=1 to +inf is converging too ? The last series can be considered as p-series or a geometric series , and it converges since n will always be positive .

    At the end does that tells me that the first series is also absolute convergent .
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  2. #2
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    Re: Check series convergence 2

    Quote Originally Posted by MirceM View Post
    I have SUM [ ((-1)^n * n!)/n^n ] from n=1 to +inf . I need to check both absolute and conditional convergence.
    \frac{{(n + 1)!}}{{{{(n + 1)}^{n + 1}}}} \cdot \frac{{{n^n}}}{{n!}} = {\left( {1 - \frac{1}{{n + 1}}} \right)^n} \to {e^{ - 1}}

    Tells us all we need to know.
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    Re: Check series convergence 2

    @Plato

    That`s the result i got too (not including e^-1) , because when n->+inf the result is 1 . I cant understand where is e^-1 from..
    Last edited by MirceM; August 24th 2014 at 09:43 AM.
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    Re: Check series convergence 2

    Quote Originally Posted by MirceM View Post
    ^ That`s the result i got too (not including e^-1) , because when n->+inf the result is 1 . I cant understand where is e^-1 from..
    Suppose that b\ne 0~\&~c\in\mathbb{Z}^+ then {\left( {1 + \frac{b}{{n + c}}} \right)^n} \to {e^b}.
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    Re: Check series convergence 2

    Yes yes .. i got it now... i hope this is the only "special case" of limits when n->inf that is a bit confusing for noobs in math like me.. Because in any other normal example i`m canceling some of the conditions and than just replace n as inf and calculate the result, like i did in this one. I mean if you dont know that term from past , you will go just str8 forward..
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