Real numbers/Series/Rules/Fact

Let

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

denote convergent series of real numbers with sums ${}s$ and ${}t$ respectively. Then the following statements hold.

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

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