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

$$\sum_{n=1}^\infty\frac{n^n+1/n}{(n+1/n)^n}$$

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  • $\begingroup$ Welcome to MathStackExchange! Please take a look at the tour page and consider editing your question to comply with the homework questions guidelines. In particular, it is good practice to give some context on what course you are taking and show your work in attempting to solve this question. $\endgroup$
    – Mefitico
    Nov 28, 2018 at 14:43

1 Answer 1

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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 $1$.

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    $\begingroup$ 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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