Leibniz Criterion

analysis

Definition

Leibniz Criterion

According to the Leibniz criterion, an alternating series ${\sum }_{n\ge 0}(-1{)}^n{a}_n$ with $a_n$ being monotonously decreasing and ${\lim }_{n\to \infty }{a}_n=0$ ($a_n$ being a zero sequence) implies that the series is convergent.

Note that this is not a classical criterion as common in mathematics since the implication does only apply in one direction.

Proof

Leibniz Criterion

Let ${\sum }_{n\ge 0}(-1{)}^n{a}_n$ be an alternating series and $s_n$ be its partial sum.

Further, let:

• $a_n$ be monotonously decreasing, and
• ${\lim }_{n\to \infty }{a}_n=0$

$$\begin{array}{rl}{s}_{2n+1}& =\underset{\ge 0}{\underbrace{(a_0-a_1)}}+\underset{\ge 0}{\underbrace{(a_2-a_3)}}+\cdots +\underset{\ge 0}{\underbrace{(a_{2n}+a_{2n+1})}}\\ s_{2n}& =a_0-\underset{\ge 0}{\underbrace{(a_1+a_2)}}-\underset{\ge 0}{\underbrace{(a_3+a_4)}}-\cdots -\underset{\ge 0}{\underbrace{(a_{2n-1}+a_{2n})}}\end{array}$$

Therefore, the sequences are bounded and monotone, and thus convergent.

$$0\le s_{2n+1}<s_{2n}\le a_0$$