Sigma notation, arithmetic series, and infinite geometric series

Sigma notation writes a sum compactly: the sum from k = 1 to 4 of (2k + 1) means 3 + 5 + 7 + 9 = 24. Sequences can be defined recursively, by giving the first term and a rule for the next, like a_n = a_(n-1) + d for an arithmetic sequence or a_n = r x a_(n-1) for a geometric one, or explicitly by an nth-term formula: a_n = a_1 + (n - 1)d, so the 10th term of 3, 7, 11, ... is 3 + 9(4) = 39, and a_n = a_1 x r^(n - 1), so the 6th term of 2, 6, 18, ... is 2 x 3^5 = 486. An arithmetic series adds the terms of an arithmetic sequence, and its sum is S_n = n(a_1 + a_n)/2, the number of terms times the average of the first and last, so 1 + 2 + ... + 100 = 100(1 + 100)/2 = 5,050. A finite geometric series has sum S_n = a_1(1 - r^n)/(1 - r). An infinite geometric series has a finite sum only when the ratio satisfies |r| < 1, and then the sum is S = a_1/(1 - r): 8 + 4 + 2 + 1 + ... = 8/(1 - 1/2) = 16. If |r| >= 1, the terms do not shrink toward 0 and the infinite series has no finite sum.

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