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:
$$\sum_{n=2}^\infty\frac{1}{n\ln n}$$
Best Answer
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.