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Sequences & Series · Series Types

Series (Partial Sums)

∑n=1∞an=lim⁡N→∞∑n=1Nan=lim⁡N→∞SN\sum_{n=1}^{\infty} a_n = \lim_{N \to \infty} \sum_{n=1}^{N} a_n = \lim_{N \to \infty} S_N

An infinite series is the limit of its partial sums. If the limit exists and is finite, the series converges.

  • 0 variables
  • 2 worked examples
  • 18 in Sequences & Series

Worked examples

Find the sum of the telescoping series Σ(1/n - 1/(n+1)) from n=1 to ∞.
  1. Sₙ = (1-1/2) + (1/2-1/3) + ... + (1/N - 1/(N+1)) = 1 - 1/(N+1)
  2. lim(N→∞) (1 - 1/(N+1)) = 1

Answer: 1

1 more worked examplePremium

Use partial sums to find Σ (1/2)ⁿ from n = 1 to ∞.

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