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Thread: determine of series converges or diverges

  1. #1
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    determine of series converges or diverges

    the sum (1+ 1/n)^2e^-n
    diverges or converges?
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  2. #2
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    Quote Originally Posted by twilightstr View Post

    the series $\displaystyle \color{red}\sum_{n=1}^{\infty} (1+ 1/n)^2e^{-n}$ diverges or converges?
    it converges because: $\displaystyle (1 + 1/n)^2 e^{-n} \leq 4e^{-n}.$
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  3. #3
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    Ok I can't really tell what the series is but I assumed it was : $\displaystyle \sum \left(1 + \frac{1}{n}\right)^{\displaystyle 2e^{-n}}$

    Use the fact that: If $\displaystyle \lim_{n \to \infty} a_n \neq 0$, then $\displaystyle \sum a_n$ diverges.

    So, raise $\displaystyle a_n$ to the power of $\displaystyle 1 = \tfrac{n}{n}$ to get:

    $\displaystyle \lim_{n \to \infty} \left[\left(1 + \frac{1}{n}\right)^{n}\right]^{\frac{2}{ne^n}} = \lim_{n \to \infty}e^{\frac{2}{ne^n}}$

    etc. etc.
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  4. #4
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    the problem is the way noncommalg wrote it.
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