CalcRef

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.

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

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