Absolute Convergence

Definition: Let $\sum a_n$ be an infinite series.
  • If the series $ {\sum |a_n|}$ converges, then we say that $\sum a_n$ is absolutely convergent.

  • If $\sum a_n$ converges but $\sum |a_n|$ doesn't, then we say that $\sum a_n$ is conditionally convergent.


Example: The alternating harmonic series $$\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} = 1 - \frac12 + \frac13 - \frac14+\ldots$$ is conditionally convergent, since the harmonic series $\sum \frac{1}{n}$ diverges.


Theorem: If $\sum a_n$ is absolutely convergent, then $\sum a_n$ converges.