Properties of Convergent Series
Theorem
If two series $\sum a_{n}$ and $\sum b_{n}$ converge, then the series $\sum c a_{n}$ ($c$ is a constant) and $\sum (a_{n} \pm b_{n})$ also converge, and the following hold.
- $\sum\limits_{n = 1}^{\infty} c a_{n} = c \sum\limits_{n = 1}^{\infty} a_{n}$
- $\sum\limits_{n = 1}^{\infty} (a_{n} \pm b_{n}) = \sum\limits_{n = 1}^{\infty} a_{n} \pm \sum\limits_{n = 1}^{\infty} b_{n}$
Explanation
This says that the constant multiple of a series and the sum of two series work out naturally, but note that it holds only for “convergent” series.
Proof
1.
$$ \begin{align*} \sum\limits_{n = 1}^{\infty} c a_{n} &= \lim\limits_{N \to \infty} \sum\limits_{n = 1}^{N} c a_{n} \\ &= \lim\limits_{N \to \infty} c \sum\limits_{n = 1}^{N} a_{n} \\ &= c \lim\limits_{N \to \infty} \sum\limits_{n = 1}^{N} a_{n} \\ &= c \sum\limits_{n = 1}^{\infty} a_{n} \\ \end{align*} $$
The first equality holds by the definition of a series, and the third equality holds by the properties of limits.
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2.
$$ \begin{align*} \sum\limits_{n = 1}^{\infty} (a_{n} \pm b_{n}) &= \lim\limits_{N \to \infty} \sum\limits_{n = 1}^{N} (a_{n} \pm b_{n}) \\ &= \lim\limits_{N \to \infty} \left(\sum\limits_{n = 1}^{N} a_{n} \pm \sum\limits_{n = 1}^{N} b_{n}\right) \\ &= \lim\limits_{N \to \infty} \sum\limits_{n = 1}^{N} a_{n} \pm \lim\limits_{N \to \infty} \sum\limits_{n = 1}^{N} b_{n} \\ &= \sum\limits_{n = 1}^{\infty} a_{n} \pm \sum\limits_{n = 1}^{\infty} b_{n} \\ \end{align*} $$
The first equality holds by the definition of a series, the second by the property of the sum $\sum$, and the third by the properties of limits.
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