Absolutely Convergent Series/Examples/Arbitrary Example 1

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
  • 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): absolutely convergent series
  • 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): absolutely convergent series
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