Math 2.0

Sequences, series, summability Definition

Partial sum

Builds on Series

Definition

Let ∑n=0∞an\sum_{n=0}^{\infty} a_n be a series of real numbers and let NN be a natural number. The NN-th partial sum of the series is the finite sum sN=∑n=0Nan.s_N = \sum_{n=0}^{N} a_n.

Remarks

The partial sums satisfy s0=a0s_0 = a_0 and sN=sN−1+aNs_N = s_{N-1} + a_N for N≥1N \ge 1. The terms are the differences of the partial sums: aN=sN−sN−1a_N = s_N - s_{N-1} for N≥1N \ge 1. For the series ∑1/2n\sum 1/2^n, the partial sums are 1,3/2,7/4,15/8,…1, 3/2, 7/4, 15/8, \dots, and sN=2−1/2Ns_N = 2 - 1/2^N.

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Sequence, definition, no evidence levelSequencedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionPartial sum, definition, no evidence levelPartial sumdefinition
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  1. Sequence definition
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194 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Series_(mathematics), Partial sum of a series
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