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Definition: Let f be a function whose domain is the positive integers and whose range is an infinite subset of the positive integers, and assume that f is one-to-one on the positive integers. Let sum(a_n) and sum(b_n) be two series such that b_n = a_(f(n)) if n is a positive integer. Then sum(b_n) is said to be a subseries of sum(a_n).
Source: Mathematical Analysis, second edition by Tom M. Apostol