# Thread: Sum of Two Series Proof

1. ## Sum of Two Series Proof

Suppose $\sum_{k=1}^{\infty} (a_k+b_k)$ converges. Then $\sum_{k=1}^{\infty} a_k$ converges <===> $\sum_{k=1}^{\infty} b_k$ converges.

Should I prove this by contradiction using partial sums of the three series?

2. What’s confusing is that there is only one case to show here (even though it looks as if not). Without loss of generality, assume one of them converges. Just look at the partial sums to get the final one.

Let $s_n=\sum_{k=1}^n(a_k+b_k)$, $r_n=\sum_{k=1}^n a_k$. Suppose $\lim s_n=s$ and $\lim r_n = r$. A fundamental limit property tells us that $\lim(s_n-r_n)=s-r$. Thus $\sum_{k=1}^nb_k$ converges.

3. Originally Posted by FailureEqualsLearn
Suppose $\sum_{k=1}^{\infty} (a_k+b_k)$ converges. Then $\sum_{k=1}^{\infty} a_k$ converges <===> $\sum_{k=1}^{\infty} b_k$ converges.

Should I prove this by contradiction using partial sums of the three series?
Hint:

Spoiler:
The sum of convergent series is convergent $x+(y-x)=y$