Sigma and sums of series

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.