(2.7.6) The Absolute Convergence Test
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.
References: