Could you tell me how to prove that if two series $ \sum_{n=0} ^{\infty}x_n, \sum_{n=0} ^{\infty} y_n$ are convergent, then $\sum_{n=0} ^{\infty}(\alpha \cdot x_n + \beta \cdot y_n)$ is also convergent and its sum equals $\alpha \cdot \sum_{n=0} ^{\infty} x_n + \beta \cdot \sum_{n=0} ^{\infty} y_n$?
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Hint: Think about the definition of $\displaystyle \sum \limits_{n=0}^{+\infty}(a_n)\color{grey}{\left(=\lim \limits_{m\to+\infty}\left(\sum \limits_{n=0}^m(a_n)\right)\right)}$ and use $\lim$ properties.
Git Gud
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This means that I can use commutitative property for $\sum_{n=0}^m (\alpha x_n + \beta y_n)= \sum_{n=0}^m \alpha x_n + \sum_{n=0}^m \beta y_n$. And then taking $\lim$ on both sides, we have the thesis. Thank you. – Hagrid Jun 12 '13 at 06:39
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@Hagrid No problem. – Git Gud Jun 12 '13 at 06:41