Formulas and Tables

Calculus

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.
  1. 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:
    \(\left| {S – {S_n}} \right| \lt \left| {{a_{n + 1}}} \right|\).
  2. 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.
  3. 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.