Leibniz Criterion

analysis

Definition

Leibniz Criterion

According to the Leibniz criterion, an alternating series converges if is a monotonically decreasing zero sequence (i.e., and ).

Note that this is a sufficient but not necessary condition; the implication only holds in one direction.

Visual Representation

The partial sums of an alternating series satisfying the Leibniz criterion “oscillate” toward the limit, with even partial sums decreasing and odd partial sums increasing.

Proof

Leibniz Criterion

Let be an alternating series and be its partial sum. Suppose is monotonically decreasing and .

We examine the subsequences of partial sums:

Since , the sequence is non-decreasing and is non-increasing. Furthermore:

Therefore, both subsequences are bounded and monotone, and thus convergent. Since , they converge to the same limit .