INFINITE SERIES

the sum of infinitely many numbers related in a given way and listed in a given order. Infinite series are useful in mathematics and in such disciplines as physics, chemistry, biology, and engineering. If a1 + a2 + a3 + . . . is an infinite series, then a quantity sn = a1 + a2 + . . . + an (n is any chosen natural number), which involves adding only finitely many terms, is called a partial sum of the series. If these numbers sn approach a fixed number S as n becomes larger and larger, the series is said to converge. In this case, S is called the sum of the series. An infinite series that does not converge is said to diverge. In the case of divergence, no value of a sum is assigned. For example, in the infinite series 1 + 1 + 1 + . . . , the nth partial sum, the result of adding the first n terms, is sn = n. As more terms are added, the partial sums fail to approach any finite value (they grow without bound). Thus, the series diverges. A basic example of a convergent series is As n becomes large, these sn approach 2, which is the sum of this infinite series. In fact, the series 1 + r + r2 + r3 + . . . converges to the sum 1/(1 - r) if 0 0 and if an + 1/an r for some r