Question

Sums and Constant Multiples of Convergent Series:

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

$\sum _{k=1}^{\infty }(a_k+b_k)=\_\_\_\_+\_\_\_\_\_$

Short answer

The required answer is$\sum _{k=1}^{\infty }(a_k+b_k)=\mathbf{\sum }_{\mathbf{k}\mathbf{=}\mathbf{1}}^{\mathbf{\infty }}\mathbf{\left(}{\mathit{a}}_{\mathbf{k}}\mathbf{\right)}\mathbf{+}\mathbf{\sum }_{\mathbf{k}\mathbf{=}\mathbf{1}}^{\mathbf{\infty }}\mathbf{\left(}{\mathit{b}}_{\mathbf{k}}\mathbf{\right)}$

Step-by-step solution

Step 1. Given Information 

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

Step 2. Explanation 

As we know that sum of two convergent series can be written as

$\sum _{k=1}^{\infty }(a_k+b_k)=\mathbf{\sum }_{\mathbf{k}\mathbf{=}\mathbf{1}}^{\mathbf{\infty }}\mathbf{\left(}{\mathit{a}}_{\mathbf{k}}\mathbf{\right)}\mathbf{+}\mathbf{\sum }_{\mathbf{k}\mathbf{=}\mathbf{1}}^{\mathbf{\infty }}\mathbf{\left(}{\mathit{b}}_{\mathbf{k}}\mathbf{\right)}$