Question

Sums and Constant Multiples of Convergent Series:

If $\sum _{k=1}^{\infty }{a}_k\mathrm{and}\sum _{k=1}^{\infty }{b}_k$ are convergent series and c is any real number, then

$\sum _{k=1}^{\infty }c{a}_k=\_\_\_\_$

Short answer

The required answer is$\sum _{k=1}^{\infty }c{a}_k=c\sum _{k=1}^{\infty }{a}_k$

Step-by-step solution

Step 1. Given Information

The given data is If $\sum _{k=1}^{\infty }{a}_k\mathrm{and}\sum _{k=1}^{\infty }{b}_k$ are convergent series and c is any real number.

Step 2. Explanation

As we know that any constant multiple of convergent series can be written as$\sum _{k=1}^{\infty }c{a}_k=\mathit{c}\mathbf{}\mathbf{\sum }_{\mathbf{k}\mathbf{=}\mathbf{1}}^{\mathbf{\infty }}{\mathit{a}}_{\mathbf{k}}$