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$(d_k)$ is a positive sequence in $\mathbb{R}$ with: $\lim_{n\to \infty} \sum_{k=1}^n d_k = \infty.$

Do the following series converge or not?

a) $\sum_{n\geq 1} \frac{d_n}{1+d_n}$ b) $\sum_{n\geq 1} \frac{ d_n}{1 + n d_n}$, c) $\sum_{n\geq1} \frac{d_n}{1 + n^2 d_n}$ d) $\sum_{n\geq 1} \frac{d_n}{1+d_n^2}$

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    You can certainly eliminate some of them by considering a constant sequence, e.g. $d_n = 1$ for all $n$. – 2017-01-11

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