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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.

  • 1 variable
  • 2 worked examples
  • 18 in Sequences & Series

Variables

Variables used in the p-Series formula
SymbolNameUnit
pExponent -

Worked examples

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

Answer: Converges. (Sum = π²/6)

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Does Σ 1/√n converge?

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Common questions

What is the p-Series?

The p-series is a fundamental test case. The harmonic series (p = 1) diverges. For p > 1, the series converges. It is one of the sequences & series formulas in the CalcRef reference.

What do the symbols in the p-Series mean?

In the p-Series, p is the exponent.

How do you use the p-Series?

Worked example. Does Σ 1/n² converge? p = 2 > 1, so it converges. Answer: Converges. (Sum = π²/6).

Practice

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