Alternating Series

Definition

An alternating series is an infinite series whose terms alternate between positive and negative values, such as $\sum (-1)^{n+1} u_n$.

• How to read: “The sum of negative one to the power of n plus one, times u n.”
• Meaning: Terms flip sign each step; the $(-1)^{n+1}$ factor creates the alternating pattern starting with a positive first term.

Why It Matters

Alternating series introduce the concept that systems with opposing forces can find stability (convergence) more easily than those with only additive parts, highlighting the subtle nature of infinite sums.

Core Concepts

• Alternating Series Test (AST): A series $\sum (-1)^{n+1} u_n$ converges if:
  1. $u_n > 0$ for all $n$.
  2. The sequence $\{u_n\}$ is non-increasing ($u_{n+1} \leq u_n$).
  3. $\lim_{n \to \infty} u_n = 0$.
  • How to read: “The sum converges if the term u n is positive, the next term u n plus one is less than or equal to u n, and the limit of u n as n approaches infinity is zero.”
  • Meaning: Three checkboxes—positive magnitudes, shrinking sizes, and terms dying to zero. If all pass, the alternating series converges.
• Remainder Estimation: For a convergent alternating series, the error $E_n$ is bounded by the magnitude of the first omitted term: $|S - s_n| \leq u_{n+1}$.
  • How to read: “The absolute value of S minus the partial sum s n is less than or equal to the next term u n plus one.”
  • Meaning: Truncation error is no worse than the next term you’d have added—simple, tight bound for alternating series.

Connected Concepts

• Conditional Convergence
• Alternating Series Estimation Theorem
• Fourier Series Convergence
• Geometric Sequence
• Geometric Series