Sigma and sums of series
Read Σ notation, find sums of arithmetic and geometric series, and see why an infinite sum can still be a number.
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Guide
A series is the sum of the terms of a sequence. The sign Σ means "add up": Σ2k for k from 1 to 5 means 2 + 4 + 6 + 8 + 10.
For an arithmetic series with first term a and common difference d, the sum of the first n terms is Sₙ = n2×(2a + (n − 1)d).
For a geometric series with first term a and common ratio r (r ≠ 1), the sum of the first n terms is Sₙ = arn − 1r − 1.
If |r| < 1 the terms shrink and shrink, and the sum to infinity is the number S = a1 − r.
Common slips: forgetting the factor in Σ; using n instead of n − 1; not halving; leaving out the "−1" in a geometric sum; multiplying by r instead of dividing by 1 − r.