Sequences & Series

Sigma notation, convergence, sum to infinity

Sigma Notation, Convergence and Sum to Infinity

Sigma notation (Σ) is a compact way to write a series. A geometric series converges (has a finite sum to infinity) only when |r| < 1. The sum to infinity is S∞ = a/(1−r).

Example

Sigma Notation

Σ (from k=1 to 5) of 2k = 2(1) + 2(2) + 2(3) + 2(4) + 2(5) = 2 + 4 + 6 + 8 + 10 = 30 Σ (from n=1 to 4) of 3ⁿ = 3 + 9 + 27 + 81 = 120

Example

Sum to Infinity

Geometric series: 8 + 4 + 2 + 1 + ... a = 8, r = 1/2. Since |r| = 1/2 < 1, series converges. S∞ = 8/(1 − 1/2) = 8/(1/2) = 16 Series: 12 − 6 + 3 − 1.5 + ... a = 12, r = −1/2. |r| < 1 → converges. S∞ = 12/(1−(−1/2)) = 12/(3/2) = 8

Note

Remember

Convergence requires |r| < 1. If |r| ≥ 1, the series diverges (sum grows without bound). Sigma notation: the variable below Σ starts at the lower limit, ends at the upper limit.

Key Vocabulary

Sigma notationThe Greek letter Σ used to represent the sum of a series

Convergent seriesA series whose sum approaches a finite value

Divergent seriesA series whose sum grows without limit

Sum to infinityS∞ = a/(1−r) for a convergent geometric series

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Sigma notation

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