Alternating Series Test (Leibniz Criterion)
Consider an alternating series $$ \sum_{k=1}^{\infty} (-1)^k b_k \quad \text{or} \quad \sum_{k=1}^{\infty} (-1)^{k+1} b_k $$ where $ {b_k} $ is a sequence of positive real numbers ( $ b_k > 0 $ ).
The series converges, meaning it approaches a finite value, provided that both of the following conditions are satisfied:
- Monotonicity: the sequence ${b_k} $ is eventually nonincreasing. In other words, from some index onward, each term is less than or equal to the previous one: $$ b_{k+1} \le b_k $$
- Limit condition: the terms of the sequence approach zero: $$ \lim_{k \to \infty} b_k = 0 $$
The Alternating Series Test, also known as the Leibniz Criterion, is one of the most direct tools for establishing the convergence of alternating series.
It applies specifically to series whose terms switch sign at each step, for example positive, negative, positive, and so on.
Note. The Alternating Series Test guarantees convergence, but it does not ensure absolute convergence. In fact, $$ \sum_{k=1}^{\infty} | (-1)^k b_k | = \sum_{k=1}^{\infty} b_k $$ may still diverge. To determine absolute convergence, additional tests are required, such as the comparison test or the ratio test.
A first example
Consider the series:
$$ \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k} $$
Here, the associated sequence is:
$$ b_k = \frac{1}{k} $$
This sequence has positive terms $ b_k > 0 $ and is decreasing, since:
$$ \frac{1}{k} \ge \frac{1}{k+1} $$
Moreover, the sequence tends to zero as $ k \to \infty $:
$$ \lim_{k \to \infty} \frac{1}{k} = 0 $$
Both conditions are satisfied, so the series converges by the Alternating Series Test.
A second example
Consider the series:
$$ \sum_{k=1}^{\infty} (-1)^{k+1} \cdot \frac{1}{\sqrt{k}} $$
In this case, the associated sequence is:
$$ b_k = \frac{1}{\sqrt{k}} $$
Again, the terms are positive and the sequence is decreasing:
$$ \frac{1}{\sqrt{k}} \ge \frac{1}{\sqrt{k+1}} $$
It also tends to zero:
$$ \lim_{k \to \infty} \frac{1}{\sqrt{k}} = 0 $$
Since both conditions are satisfied, this series converges as well.
This pattern extends to many other alternating series with similar behavior.
