# 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
• I am pretty sure there are counterexamples for both statements. If you are not satisfied with my counterexample, please provide more context so we can figure out what you would like to achieve, ultimately. – maxmilgram May 3 '19 at 17:45