Alternating Series¶

Definition¶

A series is called an alternating series if its terms change sign alternately. That is,

where $a_n>0$ for every $n\text{.}$

Interpretation¶

An alternating series is a series whose terms switch sign at every step, $a_0-a_1+a_2-\cdots$ with all $a_n>0$$0$. Consecutive terms partly cancel, so the partial sums move back and forth instead of growing monotonically.

The figure shows the partial sums $s_n$ of $1-\tfrac{1}{2}+\tfrac{1}{3}-\cdots$ for $n=1,\ldots,12$$1,\dots ,12$. They start at $s_1=1$ and $s_2=0.5$$0.5$; after that the odd-numbered sums stay above and the even-numbered sums below the limit $\ln2\approx0.693$$0.693$, closing in from both sides, with $s_{11}\approx0.737$ and $s_{12}\approx0.653$$0.653$.

In the definition, $(-1)^n$ carries the sign and $a_n$ the size of each term. A series that starts with a negative term, $-a_0+a_1-\cdots$$\cdots$, is alternating too: it is $-1$ times a series of the form above. The sign pattern alone does not give convergence: $\displaystyle \sum(-1)^n$ is alternating and diverges. The alternating series test adds the condition that $a_n$ decreases to $0$.

An alternating series can converge although $\displaystyle \sum a_n$ diverges: the alternating harmonic series in the figure converges, while the harmonic series diverges. A geometric series with a positive first term and a negative ratio is alternating, which is why, for $|q|<1$$1$, its partial sums approach the limit from alternate sides.

Key Results¶

• Alternation of signs may produce convergence through cancellation, even when the corresponding series of absolute values diverges.

Examples¶

• $\displaystyle \sum_{n=0}^\infty (-1)^n \frac{1}{n+1}$
• $\displaystyle 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots$

Prerequisites¶

• Infinite Series

Related Concepts¶

• Alternating Series Test
• Harmonic Series
• Comparison Test for Series

Sources¶

• Math I: Calculus · Lecture 10

Symbols: infinity, sum

Last updated: 20 October 2026 03:02