By Arnold Ostebee, Paul Zorn
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1 1 1 1 + − + −··· . 2! 4! 6! 8! (2) So what? Why would we write something familiar—the cosine function—in terms of something exotic—a series? One good answer is that the cosine function has no algebraic formula, and so finding accurate numerical values of cos x for arbitrary inputs x is a genuine problem. Infinite series help solve this problem. Although not quite a formula in the ordinary sense (ordinary formulas don’t include dots . . ), Equation (1) gives a concrete, computable recipe for approximating cos x: Given an input x, calculate, as far out as practically possible, the “infinite polynomial” 1− x2 x4 x6 x8 + − + −··· .
One possibility is that the partial sums Sn blow up to infinity. Another possibility is that the partial sums remain bounded but never settle on a specific limit. r Improper integrals, improper sums The definition says, in symbols, that ∞ k=1 n ak = lim n→∞ ak k=1 if the limit exists. Convergence for improper integrals means much the same thing: ∞ x=1 f (x) d x = lim n→∞ n f (x) d x x=1 if this limit exists. An infinite series is an improper sum in exactly the sense that an integral may be improper.
In short: (ak + bk ) = After all, the limit of a series is defined as the limit of the sequence of partial sums. ak + bk cak = c and ak . These reasonable-looking properties of convergent series follow directly from the analogous properties of convergent sequences. A little series algebra, cleverly applied, can immensely simplify finding the limits of certain series. ∞ E X A M P L E 6 Evaluate k=0 4 + 2k . 3k Solution The series is the sum of two (convergent) geometric series. Applying Theorems 4 and 5 gives ∞ k=0 4 + 2k = 3k ∞ k=0 ∞ 4 + 3k k=0 2 3 k = 4· 3 + 3 = 9.