Q. 85 - Calculus

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 $\sum _{k=1}^{\infty }\left(a_k+b_k\right) = L+M$.

Step-by-Step Solution

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 1. Given Information.

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

Step 2. Proof.

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

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