This question is motivated by this one. In the accepted answer, two positive non-decreasing sequences $(a_n)$ and $(b_n)$ are given such that $ \sum_{n=1}^\infty\frac{1}{a_n}=\sum_{n=1}^\infty\frac{1}{b_n}=\infty,\quad\text{but}\quad \sum_{n=1}^\infty\frac{1}{a_n+b_n}<\infty. $ Now take $b_n=n$.
Is there a positive non-decreasing sequence $(a_n)$ such that $ \sum_{n=1}^\infty\frac{1}{a_n}=\infty,\quad\text{but}\quad \sum_{n=1}^\infty\frac{1}{n+a_n}<\infty\,? $
Some remarks:
- If $a,b>0$, then $\max(a,b)\le a+b\le2\max(a,b)$. Thus $ \sum_{n=1}^\infty\frac{1}{n+a_n}<\infty\quad\text{is equivalent to}\quad\sum_{n=1}^\infty\frac{1}{\max(n,a_n)}<\infty $
- If we eliminate the condition that $(a_n)$ be non-decreasing, it is easy to find an example, like $a_n=n^2$ if $n$ is not a squere, $a_n=\sqrt{n}$ is $n$ is a square.
- If such a sequence existes, it must satisfy $ \sup\frac{a_n}{n}=\sup\frac{n}{a_n}=\infty. $