Absolutely Convergent Series/Examples/Arbitrary Example 1
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Example of Absolutely Convergent Series
Let $S$ be the series defined as:
\(\ds S\) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \paren {-1}^{n - 1} \paren {\dfrac 1 n}^n\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 - \paren {\dfrac 1 2}^2 + \paren {\dfrac 1 3}^3 - \paren {\dfrac 1 4}^4 + \cdots\) |
Then $S$ is absolutely convergent.
Proof
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Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): convergent series
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): convergent series