Let be $p$ a positive integer; $a_k$ and $b_k$ sequences of integers; $c_k$ a strictly increasing sequence of positive integers.
Suppose that
$$ p=\sum_{k=1}^{+\infty} \frac{a_k}{c_k}=\sum_{k=1}^{+\infty} \frac{b_k}{c_k}.$$
Is it sure that
$a_k=b_k$ for every $k\ge N$ for some $N$?
Let $$ a_k=\{1,0,0,0,0,0,\ldots\}\\ b_k=\{0,2,0,0,0,0,\ldots\}\\ c_k=\{1,2,3,4,5,6,\ldots\} $$