Alternating series test

In mathematical analysis, the alternating series test is the method used to prove that an alternating series with terms that decrease in absolute value is a convergent series. The test was used by Gottfried Leibniz and is sometimes known as Leibniz's test, Leibniz's rule, or the Leibniz criterion.

A series of the form

where either all a_n are positive or all a_n are negative, is called an alternating series.

The alternating series test then says: if ${\displaystyle |a_{n}|}$ decreases monotonically and ${\displaystyle \lim _{n\to \infty }a_{n}=0}$ then the alternating series converges.

Moreover, let L denote the sum of the series, then the partial sum

...
Wikipedia