Math 2.0

Sequences, series, summability Series: basic notions Concept

Telescoping series

Builds on SeriesPartial sum

Summary

A telescoping series is a series whose terms have the form an=bn−bn+1a_n = b_n - b_{n+1} for a sequence (bn)(b_n). Its partial sums collapse: ∑n=0N(bn−bn+1)=b0−bN+1\sum_{n=0}^{N} (b_n - b_{n+1}) = b_0 - b_{N+1}. Thus the series converges if and only if (bn)(b_n) converges, and then its sum is b0−lim⁡bnb_0 - \lim b_n.

Remarks

For example, 1n(n+1)=1n−1n+1\frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1}, so ∑n=1∞1n(n+1)=1\sum_{n=1}^{\infty} \frac{1}{n(n+1)} = 1. The finite form ∑k=0n−1(f(k+1)−f(k))=f(n)−f(0)\sum_{k=0}^{n-1} (f(k+1) - f(k)) = f(n) - f(0) is a discrete analogue of the fundamental theorem of calculus. Mathlib states it as Finset.sum_range_sub.

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Sequence, definition, no evidence levelSequencedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionPartial sum, definition, no evidence levelPartial sumdefinitionTelescoping series, concept, no evidence levelTelescoping seriesconcept
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  1. Sequence definition
  2. Real sequence definition
  3. Series definition
  4. Partial sum definition
  5. Telescoping series concept

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2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Telescoping_series, Definition
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