Real numbers/Series/Rules/Fact/Proof/Exercise

Let

$\sum _{k=0}^{\infty }a_{k}{\text{ and }}\sum _{k=0}^{\infty }b_{k}$

be convergent series of real numbers with sums ${}s$ and ${}t$. Prove the following statements.

1. The series ${}\sum _{k=0}^{\infty }c_{k}$ with ${}c_{k}=a_{k}+b_{k}$ is convergent with sum equal to ${}s+t$.
2. For ${}r\in \mathbb {R}$ the series ${}\sum _{k=0}^{\infty }d_{k}$ with ${}d_{k}=ra_{k}$ is also convergent with sum equal to ${}rs$.

Create a solution