# Sequence identity

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 $$a_k=\{1,0,0,0,0,0,\ldots\}\\ b_k=\{0,2,0,0,0,0,\ldots\}\\ c_k=\{1,2,3,4,5,6,\ldots\}$$
• and if $a_k$ and $b_k$ are not $0$ definitely? – Saverio Picozzi May 3 '19 at 12:37
• or is it true that $a_k=b_k$ for every $k \ge N$ for some $N$ ? – Saverio Picozzi May 3 '19 at 12:42