Prove the shift rule for series

Question

Let $N$ be a natural number. Then the series $\sum_{n=1}^{\infty}a_n$ converges if and only if the series $\sum_{n=1}^{\infty}a_{N+n}$ converges.

I've tried splitting $\sum_{n=1}^{\infty}a_n$ into $\sum_{n=1}^{\infty}a_{N+n} +\sum_{n=1}^{N}a_n$ but am not getting anywhere. I've also tried using the $\epsilon$ definition for convergence but am getting confused.

Answer 1

$\sum_{n=1}^Na_n$ is just a number, let's call it $C$. So if the sum of $a_n$ converges to $A$, then the $$A=\sum_{n=1}^\infty a_n=\sum_{n=1}^N a_n+\sum_{n=1}^\infty a_{n+N}=C+\sum_{n=1}^\infty a_{n+N}$$ wich yields $$\sum_{n=1}^\infty a_{n+N}=A-C$$ Since both $A$ and $C$ are finite then $A-C$ is finite, so the series converges.

Similarly, you can go the other way around

Answer 2

As you have written we have $$\sum_{n=1}^{\infty}a_n=\sum_{n=1}^{\infty}a_{N+n} +\sum_{n=1}^{N}a_n$$where $$\sum_{n=1}^{N}a_n$$ is bounded therefore $\sum_{n=1}^{\infty}a_n$ is bounded if and only if $\sum_{n=1}^{\infty}a_{n+N}$ is bounded.