Is this series convergent or divergent ? (The sum will be infinite or it will converge to a certain number?)


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    – Mefitico
    Nov 28, 2018 at 14:43

1 Answer 1


You can certainly ignore the term $\dfrac1n$ at the numerator and

$$\sum\frac{n^n}{(n+\frac1n)^n}=\sum\frac1{(1+\frac1{n^2})^n}$$ diverges as the general term tends to $1$.

  • 1
    $\begingroup$ At first, I lost the n^n term at the numerator. $\endgroup$
    – user
    Nov 28, 2018 at 14:52
  • $\begingroup$ Shouldn’t we use one of the series tests to answer this question? $\endgroup$ Nov 28, 2018 at 15:12
  • $\begingroup$ @Negarrezaei: it is enough to note that as $(1+1/n^2)^n=\sqrt[n]{(1+1/n^2)^{n^2}}$, the general term goes to $1$. $\endgroup$
    – user65203
    Nov 28, 2018 at 15:15
  • $\begingroup$ @YvesDaoust even befor simplifying , the sum of the very first 3 characters of the series would be 2.45. How can the sum of the whole characters of the series from 1 to infinitive would be convergent to 1 ? $\endgroup$ Nov 28, 2018 at 15:39
  • $\begingroup$ @Negar: I am talking about the general term, not the partial sums. And I said that the sum diverges !!! $\endgroup$
    – user65203
    Nov 28, 2018 at 16:00

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