Question

Prove that if $\sum _{k=1}^{\infty }{a}_k$converges to L and $\sum _{k=1}^{\infty }{b}_k$converges to M , then the series$\sum _{k=1}^{\infty }\left(a_k+b_k\right)=L+M$.

Short answer

$$\begin{gathered} \sum _{k=1}^{\infty }\left(a_k+b_k\right)=\sum _{k=1}^{\infty }{a}_k+\sum _{k=1}^{\infty }{b}_k=L+M. \\ Hence,proved. \end{gathered}$$

Step-by-step solution

Step 1. Given Information.

$\sum _{k=1}^{\infty }{a}_k=Land\sum _{k=1}^{\infty }{b}_k=M.$

Step 2. Proof.

We know that the sum of two convergent series is a convergent series itself.

$$\begin{gathered} \sum _{k=1}^{\infty }{a}_k=L,and\sum _{k=1}^{\infty }{b}_k=M. \\ So, \\ \sum _{k=1}^{\infty }\left(a_k+b_k\right)=\sum _{k=1}^{\infty }{a}_k+\sum _{k=1}^{\infty }{b}_k \\ =L+M. \\ Hence,proved. \end{gathered}$$