Definition

If the series \(\sum_{n=1}^{\infty}|a_n|\) converges, then \(\sum_{n=1}^{\infty} a_n\) converges as well.


Proof

Given the series \(\sum_{n=1}^{\infty}|a_n|\) converges, we know that for any \(\epsilon > 0\), there must exist a number \(N \in \mathbf{N}\) such that whenever \(n > m \geq N\),

$$ \begin{align*} \left| |a_{m+1}| + |a_{m+2}| + ... + |a_{n}| \right| &= |a_{m+1}| + |a_{m+2}| + ... + |a_{n}| \quad \text{ because $|a_k| \geq 0$ for any $k$}\\ &< \epsilon. \end{align*} $$

By the triangle inequality, we know that

$$ \begin{align*} |a_{m+1} + a_{m+2} + ... + a_{n}| \leq |a_{m+1}| + |a_{m+2}| + ... + |a_{n}|. \end{align*} $$

Therefore, \(|a_{m+1} + a_{m+2} + ... + a_{n}| < \epsilon\) and so by the Cauchy criterion for series, \(\sum_{n=1}^{\infty}a_n\) also converges.

Other Definitions and Properties

  • Fo the absolute value function definition and other properties, see here.
  • For the definitions of sequences and what it means to for a sequence to converge, see this, for subsequences see this.
  • For the definitions of series, and what it means to for a series to converge, see this.
  • For the "show the limit" template and an example, see this.


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