Andromeda
Note

The nth-Term Test

Definition

An infinite series is the formal sum of the terms of an infinite sequence: ∑n=1∞an=a1+a2+a3+…\sum_{n=1}^{\infty} a_n = a_1 + a_2 + a_3 + \dots. Its value is defined as the limit of the sequence of its partial sums sns_n, where sn=∑k=1naks_n = \sum_{k=1}^{n} a_k.

  • How to read: “The sum from n equals one to infinity of a n, where the partial sum s n is defined as the sum from k equals one to n of a k.”
  • Meaning: An infinite series is the limit of its finite partial sums — convergence means partial sums approach a finite total.

Why It Matters

This test is the ultimate “filter” for saving time and mental energy. In the world of infinite series, most attempts to find a sum will fail (diverge). This test allows you to quickly identify the “impossible” cases before you waste hours on a problem that has no solution. It is a fundamental lesson in identifying the necessary conditions for success before worrying about the sufficient ones.

Core Concepts

  • Partial Sums: The sequence sns_n tracks the cumulative total. The series converges if lim⁡n→∞sn\lim_{n \to \infty} s_n exists.

    • How to read: “The sequence s n converges if the limit of s n as n approaches infinity exists.”
    • Meaning: The series sum is defined as the limit of partial sums — convergence of {sn}\{s_n\} defines convergence of the series.

  • nnth-Term Test for Divergence: If lim⁡n→∞an≠0\lim_{n \to \infty} a_n \neq 0 (or the limit doesn’t exist), the series must diverge. This is a “sanity check” test; it can prove divergence but never proves convergence.

    • How to read: “If the limit of a n as n approaches infinity is not zero, the series diverges.”
    • Meaning: If terms don’t shrink to zero, you keep adding roughly the same amount forever — divergence is guaranteed. Cannot prove convergence.

  • Geometric Series: A specific series ∑arn\sum ar^n that converges to a1−r\frac{a}{1-r} if and only if ∣r∣<1|r| < 1.

    • How to read: “The sum of a times r to the n converges to a divided by the quantity one minus r, provided the absolute value of r is less than one.”
    • Meaning: Each term is a fixed fraction of the previous; converges when ∣r∣<1|r| < 1 because terms decay exponentially.

  • Linearity: Series can be added, subtracted, and multiplied by constants, provided they converge.

Connected Concepts

Connected notes