Sum of Absolutely Convergent Series
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Theorem
Let $\ds \sum_{n \mathop = 1}^\infty a_n$ and $\ds \sum_{n \mathop = 1}^\infty b_n$ be two real or complex series that are absolutely convergent.
Then the series $\ds \sum_{n \mathop = 1}^\infty \paren {a_n + b_n}$ is absolutely convergent, and:
- $\ds \sum_{n \mathop = 1}^\infty \paren {a_n + b_n} = \sum_{n \mathop = 1}^\infty a_n + \sum_{n \mathop = 1}^\infty b_n$
Proof
Let $\epsilon \in \R_{>0}$.
From Tail of Convergent Series tends to Zero, it follows that there exists $M \in \N$ such that:
- $\ds \sum_{n \mathop = M + 1}^\infty \cmod {a_n} < \dfrac \epsilon 2$
and:
- $\ds\sum_{n \mathop = M + 1}^\infty \cmod {b_n} < \dfrac \epsilon 2$
For all $m \ge M$, it follows that:
\(\ds \cmod {\sum_{n \mathop = 1}^\infty a_n + \sum_{n \mathop = 1}^\infty b_n - \sum_{n \mathop = 1}^m \paren {a_n + b_n} }\) | \(=\) | \(\ds \cmod {\sum_{n \mathop = m + 1}^\infty a_n + \sum_{n \mathop = m + 1}^\infty b_n}\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \sum_{n \mathop = m + 1}^\infty \cmod {a_n} + \sum_{n \mathop = m + 1}^\infty \cmod {b_n}\) | by Triangle Inequality | |||||||||||
\(\ds \) | \(\le\) | \(\ds \sum_{n \mathop = M + 1}^\infty \cmod {a_n} + \sum_{n \mathop = M + 1}^\infty \cmod {b_n}\) | ||||||||||||
\(\ds \) | \(<\) | \(\ds \epsilon\) |
By definition of convergent series, it follows that:
\(\ds \sum_{n \mathop = 1}^\infty a_n + \sum_{n \mathop = 1}^\infty b_n\) | \(=\) | \(\ds \lim_{m \mathop \to \infty} \sum_{n \mathop = 1}^m \paren {a_n + b_n}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \paren {a_n + b_n}\) |
To show that $\ds \sum_{n \mathop = 1}^\infty \paren {a_n + b_n}$ is absolutely convergent, note that:
\(\ds \sum_{n \mathop = 1}^\infty \cmod {a_n} + \sum_{n \mathop = 1}^\infty \cmod {b_n}\) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \paren {\cmod {a_n} + \cmod {b_n} }\) | as shown above | |||||||||||
\(\ds \) | \(\ge\) | \(\ds \sum_{n \mathop = 1}^\infty \cmod {a_n + b_n}\) | by Triangle Inequality |
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.4$. Power Series
- 1992: Larry C. Andrews: Special Functions of Mathematics for Engineers (2nd ed.) ... (previous) ... (next): $\S 1.2.3$: Operations with series