# Properties of Alternating Series

• Number sequence: $$\left\{ {a_n} \right\}$$
Alternating series: $$\sum\limits_{n = 1}^\infty {{{\left( { – 1} \right)}^n}{a_n}} ,$$ $$\sum\limits_{n = 1}^\infty {{{\left( { – 1} \right)}^{n – 1}}{a_n}}$$
Number of terms in a series: $$n$$
Partial sum of a series: $${S_n}$$
Sum of a series: $$S$$
1. An infinite series in which successive terms have opposite signs is called an alternating series. An alternating series can be written in the form
$$\sum\limits_{n = 1}^\infty {{{\left( { – 1} \right)}^n}{a_n}}$$ or $$\sum\limits_{n = 1}^\infty {{{\left( { – 1} \right)}^{n – 1}}{a_n}}.$$
2. Alternating series test (Leibniz’s theorem)
The alternating series test is a sufficient condition for convergence of an alternating series. Let $$\left\{ {a_n} \right\}$$ be a sequence of positive terms such that
$$-\;{a_{n + 1}} \lt {a_n}$$ for all $$n$$;
$$-\;\lim\limits_{n \to \infty } {a_n} = 0$$.
Then the alternating series $$\sum\limits_{n = 1}^\infty {{{\left( { – 1} \right)}^n}{a_n}}$$ and $$\sum\limits_{n = 1}^\infty {{{\left( { – 1} \right)}^{n – 1}}{a_n}}$$ both converge.
3. Alternating series remainder estimate
Suppose that an alternating series converges by the alternating series test and its sum is equal to $$S$$. We denote the $$n$$th partial sum of the series as $${S_n}.$$ Then the remainder of the alternating series in absolute value is bounded by the absolute value of the first discarded term:
$$\left| {S – {S_n}} \right| \lt \left| {{a_{n + 1}}} \right|$$.
4. Absolute convergence
A series $$\sum\limits_{n = 1}^\infty {{a_n}}$$ is called absolutely convergent if the series $$\sum\limits_{n = 1}^\infty {\left| {{a_n}} \right|}$$ composed of the absolute values of the terms $${a_n}$$ is convergent. If the series $$\sum\limits_{n = 1}^\infty {{a_n}}$$ is absolutely convergent, then it is (just) convergent. The converse is not generally true.
5. Conditional convergence
A series $$\sum\limits_{n = 1}^\infty {{a_n}}$$ is called conditionally convergent if it converges, but the series $$\sum\limits_{n = 1}^\infty {\left| {{a_n}} \right|}$$ composed of the absolute values of the terms $${a_n}$$ diverges. In other words, a series is conditionally convergent if it converges, but not absolutely.