CalcRef

Sequences & Series · Series Types

p-Series

∑n=1∞1np converges if p>1, diverges if p≤1\sum_{n=1}^{\infty} \frac{1}{n^p} \text{ converges if } p > 1 \text{, diverges if } p \leq 1

The p-series is a fundamental test case. The harmonic series (p = 1) diverges. For p > 1, the series converges.

Variables

SymbolNameUnit
pExponent—

Worked examples

Does Σ 1/n² converge?
  1. p = 2 > 1, so it converges.

Answer: Converges. (Sum = π²/6)

Does Σ 1/√n converge?
  1. 1/√n = 1/n^(1/2). p = 1/2 ≤ 1, so it diverges.

Answer: Diverges.

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