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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.

  • 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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Common questions

What is the Series (Partial Sums)?

An infinite series is the limit of its partial sums. If the limit exists and is finite, the series converges. It is one of the sequences & series formulas in the CalcRef reference.

How do you use the Series (Partial Sums)?

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

Practice

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