Absolute and Conditional Convergence of Series of Real Numbers

We will now classify two very special and different types of convergent of series of real numbers which we define below.

Definition: A series $\displaystyle{\sum_{n=1}^{\infty} a_n}$ is said to be Absolutely Convergent if $\displaystyle{\sum_{n=1}^{\infty} \mid a_n \mid}$. Furthermore, a convergent series $\displaystyle{\sum_{n=1}^{\infty} a_n}$ is said to be Conditionally Convergent if $\displaystyle{\sum_{n=1}^{\infty} \mid a_n \mid}$ diverges.

In the definition above for absolute convergence we did not require that $\displaystyle{\displaystyle{\sum_{n=1}^{\infty} a_n}}$ needed to be convergent. In fact, if $\displaystyle{\sum_{n=1}^{\infty} \mid a_n \mid}$ then we can immediately conclude that $\displaystyle{\sum_{n=1}^{\infty} a_n}$ converges as we prove in the following theorem.

Theorem 1: Every absolutely convergent series is convergent, that is, if $\displaystyle{\sum_{n=1}^{\infty} \mid a_n \mid}$ converges then $\displaystyle{\sum_{n=1}^{\infty} a_n}$ converges.

Be cautious. The converse of Theorem 1 is not true in general.

• Proof: Suppose that $\displaystyle{\sum_{n=1}^{\infty} \mid a_n \mid}$ converges. The by Cauchy's Condition for Convergent Series we have that for all $\epsilon > 0$ there exists an $N \in \mathbb{N}$ such that if $n \geq N$ we have that for all $p = 1, 2, ...$ that:

(1)

\begin{align} \quad \mid \mid a_{n+1} \mid + \mid a_{n+2} \mid + ... + \mid a_{n + p} \mid \mid < \epsilon \end{align}

• But then for all $n \geq N$ we have that for all $p = 1, 2, ...$ that:

(2)

\begin{align} \quad \mid a_{n+1} + a_{n+2} + ... + a_{n+p} \mid \leq \mid \mid a_{n+1} \mid + \mid a_{n+2} \mid + ... + \mid a_{n + p} \mid \mid < \epsilon \end{align}

• So Cauchy's condition is satisfied for the series $\displaystyle{\sum_{n=1}^{\infty} a_n}$ which shows that $\displaystyle{\sum_{n=1}^{\infty} a_n}$ converges. $\blacksquare$