Slowing down divergence 2 Let $f(x)$ and $g(x)$ be positive nondecreasing functions such that
$
\sum_{n&gt;1} \frac1{f(n)} \text{ and } \sum_{n&gt;1} \frac1{g(n)}
$

diverges.  

(Why) must the series $$\sum_{n&gt;1} \frac1{g(n)+f(n)}$$ diverge?
No, $\sum\frac{1}{f(n)+g(n)}$ need not diverge.  We define such a pair $f,g$ inductively.  Let $f(1) = g(1) = 2$.

Suppose $f(n)$ and $g(n)$ have been defined for $1\leq n\leq N$.  Let $L$ be the largest value taken by $f(n)$ or $g(n)$ for $1\leq n\leq N$.  Define $f(N+1) = f(N+2) = \cdots = f(N + L) = L$ and $g(N+i) = 2^iL$ for $1\leq i\leq L$.  Define $g(N+L+1) = \cdots = g(N+L+2^LL) = 2^LL$ and $f(N+L+i) = 2^{L+i}L$ for $i\leq 1\leq 2^LL$.  The extends the domain of definition of $f(n)$ and $g(n)$ to $1\leq n\leq N+L+2^LL$.  Repeat to define $f$ and $g$ on all of $\mathbb{N}$.

Note that $f$ and $g$ are positive and nondecreasing.  In each stage of this process there are $L$ times when $f$ takes the value $L$ and $2^LL$ times when $g$ takes the value $2^LL$.  Therefore $\sum_{n=N+1}^{N+L+2^LL} \frac{1}{f(n)} &gt; 1$ and $\sum_{n=N+1}^{N+L+2^LL} \frac{1}{g(n)} &gt; 1$: at least $1$ is added to the sum of the reciprocals of each series at each step.  Since we repeat this process ad infinitum, adding at least $1$ to each series at each step,  $\sum_{n=1}^\infty \frac{1}{f(n)}$ and $\sum_{n=1}^\infty \frac{1}{g(n)}$ both diverge.

Note that by construction, $\max(f(n+1),g(n+1)) = 2\max(f(n),g(n))$ for all $n$.  Since $f(1)=g(1)=2$, we have $\max(f(n),g(n))=2^n$ for all $n$.   Therefore $\frac{1}{f(n)+g(n)}\leq \frac{1}{\max(f(n),g(n))} = 2^{-n}$ for all $n$, so $\sum_{n=1}^\infty\frac{1}{f(n)+g(n)}$ converges by comparison with the geometric series $2^{-n}$.
This is a follow-up to Gortaur's "answer" (and isn't a real answer either).

If $a_n$ and $c_n$ are as above, then to show that $\sum_nc_n=\infty$, I believe that it is sufficient to show that 
if $\;1=i_0&lt; i_1&lt;i_2&lt;\dots\;$ is any infinite sequence of indexes, then $\sum_jd_j=\infty$ where 
$$
d_j\;=\;a_{i_k} \;\;\textrm{where}\;\; i_{k-1}\leqslant j&lt;i_k.
$$
This reduces the question to one about a single non-increasing positive sequence. I haven't been able to find a counter-example to this.