Q. 84 - Calculus

Question

Prove Theorem 7.24 (a). That is, show that if c is a real number and $\sum _{k=1}^{\infty }{a}_k$ is a convergent series, then $\sum _{k=1}^{\infty }c{a}_k=c\sum _{k=1}^{\infty }{a}_k$.

Step-by-Step Solution

Answer

As $\sum _{k=1}^{\infty }{a}_k$ is a convergent series, and c is constant we get c out of the summation and we prove that $\sum _{k=1}^{\infty }c{a}_k=c\sum _{k=1}^{\infty }{a}_k$.

Step 1. Given Information.

We are given that $\sum _{k=1}^{\infty }{a}_k$ is a convergent series and c is a real number. 

We need to show that  $\sum _{k=1}^{\infty }c{a}_k=c\sum _{k=1}^{\infty }{a}_k$.

Step 2. Proof.

The series $\sum _{k=1}^{\infty }c{a}_k$ can be written in expanded form as

$\sum _{k=1}^{\infty }c{a}_k=c{a}_1+c{a}_2+...$

It can be factorized and written as

$$\begin{gathered} \sum _{k=1}^{\infty }c{a}_k=c{a}_1+c{a}_2+... \\ \sum _{k=1}^{\infty }c{a}_k=c(a_1+a_2+...) \\ \sum _{k=1}^{\infty }c{a}_k=c\sum _{k=1}^{\infty }{a}_k \end{gathered}$$

Thus, for a real number c it can be shown that $\sum _{k=1}^{\infty }c{a}_k=c\sum _{k=1}^{\infty }{a}_k$.