I have to find out, whether this sum converges:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

\sum_{n = 2}^{+ \infty} n^{\alpha} \left( \log (n+1) - \log n \right)^4

[/tex]

So I rewrote it:

[tex]

\sum_{n = 2}^{+ \infty} n^{\alpha} \left( \log \left(1 + \frac{1}{n} \right) \right)^4 =

[/tex]

[tex]

\sum_{n = 2}^{+ \infty} n^{\alpha - 4} \left[ \frac {\log \left(1 + \frac{1}{n} \right)}{\frac{1}{n}} \right]^4

[/tex]

Now I can see that the expression with log goes to 1, so the sum gets reduced to

[tex]

\sum_{n = 2}^{+ \infty} \frac {1}{n^{4 - \alpha}}

[/tex]

and the result is obviously [itex]\alpha \in \left(0, 3\right)[/itex].

But, I don't know how exactly should I justify my decision. I'm not talking about the very last step now, but the one in which I threw away the logarithm because of the limit 1. We have to give reasons for each non-trivial step we do, so I think I should write some theorem or rule under that step...

Thank you.

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# Homework Help: Sum convergence - is my approach flawless?

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