Absolute and Conditional Convergence Examples 1

Recall from the Absolute and Conditional Convergence page that if $\sum_{n=1}^{\infty} a_n$ is a convergent series, then we further say that $\sum_{n=1}^{\infty} a_n$ is absolutely convergent if $\sum_{n=1}^{\infty} \mid a_n \mid$ converges. If $\sum_{n=1}^{\infty} \mid a_n \mid$ diverges, then we say that the original series is conditionally convergent.

Furthermore, we say that if $\sum_{n=1}^{\infty} \mid a_n \mid$ was convergent, then $\sum_{n=1}^{\infty} a_n$ was convergent.

We will now look at some examples regarding absolute and conditional convergence of general series.

Example 1

Suppose that $\sum_{n=1}^{\infty} a_n$ converges. Must $\sum_{n=1}^{\infty} (-1)^n a_n$ converge as well?

Consider the series $\sum_{n=1}^{\infty} \frac{(-1)^n}{n}$. We will see later with The Alternating Series Test that this series converges (it was briefly mentioned that this series converges on the Absolute and Conditional Convergence page).

We then have that:

(1)

\begin{align} \quad \sum_{n=1}^{\infty} (-1)^n a_n = \sum_{n=1}^{\infty} (-1)^n \frac{(-1)^n}{n} = \sum_{n=1}^{\infty} \frac{1}{n} \end{align}

But this series diverges as the harmonic series.

Example 2

Suppose that $\sum_{n=1}^{\infty} a_n$ converges absolutely. Prove that then $\sum_{n=1}^{\infty} (-1)^n a_n$ converges absolutely.

Consider the series $\sum_{n=1}^{\infty} (-1)^n a_n$. Take the absolute value of the terms to get the series:

(2)

\begin{align} \quad \sum_{n=1}^{\infty} \mid (-1)^n a_n \mid = \sum_{n=1}^{\infty} \mid a_n \mid \end{align}

Since $\sum_{n=1}^{\infty} a_n$ converges absolutely, we have that $\sum_{n=1}^{\infty} \mid a_n \mid$ converges, and so the series above converges. Therefore $\sum_{n=1}^{\infty} (-1)^n a_n$ converges absolutely.