Question

Let $\sum _{k=1}^{\infty }{a}_{k}beaconvergentseriesand\sum _{k=1}^{\infty }{b}_{k}beadivergentseries.$Prove that the series $\sum _{k=1}^{\infty }\left(a_k+b_k\right)$diverges.

Short answer

Proof by method of contradiction.

$\sum _{k=1}^{\infty }\left(a_k+b_k\right)$ is a divergent series.

Step-by-step solution

Step 1. Given Information.

$\sum _{k=1}^{\infty }{a}_k$is a convergent series and $\sum _{k=1}^{\infty }{b}_k$ is a divergent series.

Step 2. Proof by method of contradiction.

Let suppose $\sum _{k=1}^{\infty }\left(a_k+b_k\right)$be a convergent series.

Now, we know the sum of two convergent series is a convergent series.

So, $\sum _{k=1}^{\infty }\left(a_k+b_k\right)-\sum _{k=1}^{\infty }\left(a_k\right)$will also be convergent.

Which can be implied as $\sum _{k=1}^{\infty }\left(a_k+b_k-a_k\right)=\sum _{k=1}^{\infty }\left(b_k\right)$must be convergent.

But it is given that $\sum _{k=1}^{\infty }\left(b_k\right)$is a divergent series.

So it contradicts and so our assumption is wrong.

Thus the series $\sum _{k=1}^{\infty }\left(a_k+b_k\right)$is a divergent series.