📂Calculus

Absolute Convergence and Conditional Convergence of Series

Definition¹

For a series $\sum\limits_{n=0}^{\infty} a_{n}$, if $\sum\limits_{n=0}^{\infty} |a_{n}|$ converges, then $\sum\limits_{n=0}^{\infty} a_{n}$ is said to converge absolutely.

Explanation

The point to note is that a given series is said to converge absolutely not when the given series itself converges, but when “the series obtained by taking the absolute value of each term of the series” converges. This means that a convergent series may not converge absolutely. Considering the alternating harmonic series, this series converges, but the harmonic series does not converge. In such a case, the alternating harmonic series is said to converge conditionally.

Conditional Convergence

When the series $\sum\limits_{n=0}^{\infty} a_{n}$ converges but $\sum\limits_{n=0}^{\infty} |a_{n}|$ diverges, $\sum\limits_{n=0}^{\infty} a_{n}$ is said to converge conditionally.

Theorem

If the series $\sum a_{n}$ converges absolutely, then it converges.

Proof

The inequality $0 \le a_{n} + |a_{n}| \le 2 |a_{n}|$ holds. Since $\sum a_{n}$ converges absolutely, $\sum 2 |a_{n}|$ also converges. Then, by the comparison test, $\sum a_{n} + |a_{n}|$ also converges. Therefore, $\sum a_{n}$ converges, since it is the difference of two convergent series as shown below.

$$ \sum a_{n} = \sum (a_{n} + |a_{n}|) - \sum |a_{n}| \lt \infty $$

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