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

To I am trying to figure out exactly what to compare this series to in order to prove that it diverges. I know that $1/n$ diverges and I also know that by the p-series the square root diverges as well. Can anybody give me a good way to approach this problem? Thanks!

Zev Chonoles
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EhBabay
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1 Answers1

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HINT

Compare with $\displaystyle \sum_{n=2}^{\infty} \dfrac1{n \ln(n)}$ and conclude from here.

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    How can you compare that sum when at 1 the sum is undefined? – EhBabay Apr 08 '13 at 03:52
  • @JohnCarpenter Ok. Start from $2$. –  Apr 08 '13 at 03:53
  • Ok so how can you compare two sums that have diff starting points? – EhBabay Apr 08 '13 at 03:53
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    @JohnCarpenter Write your series as the first term + the terms starting from $n=2$. And note that it converges/diverges only when the series from $n=2$ converges/diverges respectively. –  Apr 08 '13 at 03:55
  • Ah thanks you sir. – EhBabay Apr 08 '13 at 03:55
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    @JohnCarpenter: The important point is that convergence only depends on what happens "at infinity". Anything that happens "down low" can change the limit, but can't change whether the series converges, so can be ignored. A few extra terms, some (finite) number of terms altered, etc, don't change whether the series converges. – Ross Millikan Apr 08 '13 at 04:04