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When testing to determine the convergence or divergence of series with positive terms, there's a common way by comparing them with other series which we already know converge or diverge.

My question is, how do we choose the proper to-be-compared series? I hope to get some detailed methodology about this. I am a bit confused - do I have to even rely on my intuition?

For instance, how do I choose a comparison series for this given one below:

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

2 Answers 2

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In your case, the convergence of $\displaystyle \sum_{n=2}^{\infty} \dfrac1{n \log n}$ can be checked by using the following convergence test. If we have a monotone decreasing sequence, then $\displaystyle \sum_{n=2}^{\infty} a_n$ converges iff $\displaystyle \sum_{n=2}^{\infty} 2^na_{2^n}$ converges.

Note that $\sum_{n=2^k}^{2^{k+1}-1}\dfrac1{n \log n} > \sum_{n=2^k}^{2^{k+1}-1} \dfrac1{2^k \log \left(2^k \right)} = \dfrac{2^k}{2^k k \log(2)} = \dfrac1{\log 2} \dfrac1k$

Hence, $\displaystyle \sum_{n=2}^{\infty} \dfrac1{n \log n} > \dfrac1{\log 2} \sum_{k=1}^{\infty} \dfrac1k$ Hence, it diverges.

In general, if you want to prove $\sum_{k=1}^{\infty} a_k$ diverges and you are unable to find $b_k$ such that $a_k > b_k$ and $\sum_{k=1}^{\infty} b_k$ diverges, your next bet is to find $b_k$ such that $\displaystyle \sum_{n=f(k)}^{f(k+1)-1} a_n > b_k$, where $f(k)$ is some strictly monotone increasing function, such that $\sum b_k$ diverges.

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    I know marty, I just had a question about Marvis's answer. @Marvis Thanks!2012-11-04
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Usually, you can choose the highest order of the denominator and numerator, respectively. So in your problem, for the numerator choose $1$, whose highest order is $0$, and for denominator choose $n$. Then, the sequence you should choose is $1/n$.

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    it seems that the 1/n can't work out here.2012-11-04