# For an,bn↑a_n,b_n\uparrow and ∑1an\sum \frac{1}{a_n}, ∑1bn\sum \frac{1}{b_n} divergent is the series ∑1an+bn\sum \frac{1}{a_n+b_n} also divergent? [duplicate]

Let $a_n$ and $b_n$ are strictly increasing to $+\infty$ sequences such that the series $\sum \frac{1}{a_n}$ and $\sum \frac{1}{b_n}$ are divergent. Is it true that the series $\sum \frac{1}{a_n+b_n}$ is also divergent?

At first sight it looks true, so I tried to prove that, but after some failed attempts I start to believe now that there might exist a counterexample. Any thoughts?

For $r \geqslant 0$, let $k_r = 2^{2^r}$. Let

for $n \geqslant 4$, and choose $a_n, b_n$ fairly arbitrarily for $n < 4$.

Then

so $\sum \frac{1}{a_n}$ diverges. Analogously, $\sum \frac{1}{b_n}$ diverges.

But, we have $a_n > b_n$ for $k_{2r} \leqslant n < k_{2r+1}$, and $b_n > a_n$ for $k_{2r+1} \leqslant n < k_{2r+2}$, so

and

so

and

converges.