Does series converge or diverge

Asked May 26 '23 at 17:02

Active May 26 '23 at 17:02

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I need to figure out if the following series converges or diverges:

$\sum_{n=1}^{\infty} (\frac{n}{n+2})^{(n(n+1))}$

I tried using Cauchy's criterion:

$C_n = (\frac{n}{n+2})^{n+1}$

It seems to be less than 1 so I thought the series should converge but then realized that if I apply this method for example to these series $\sum_{n=1}^{\infty} (\frac{1}{n})$, then it will also be less than one but the series diverges.

So what would be the correct approach here?

asked May 26 '23 at 17:02

Alisa Petrova

You are not applying Cauchy's root test properly. Look at $\lim C_n,$ not just at $C_n$ (For the harmonic series, the limit is $1^-$ so the criterion is inconclusive). – Anne Bauval May 26 '23 at 17:06
Possible suggestion: You've basically set up for the root test, now do long division on $\frac{n}{n+2}$ and a familiar limit should appear. – Sean Roberson May 26 '23 at 17:07
Or simply: $\frac1{C_n}\sim\left(1+\frac2n\right)^n.$ – Anne Bauval May 26 '23 at 17:20
3

Does this answer your question? Evaluate $\lim\limits_{x \to { \infty } } (\frac{x}{x+2})^x$. Need for explanation. Together with the previous comment, it proves that $\lim C_n=e^{-2}<1.$ – Anne Bauval May 26 '23 at 17:23

Does series converge or diverge

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