In short power series with smaller coecients have

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Unformatted text preview: ge for any nonzero x. The ratio test doesn’t always work because the limit may not exist, but sometimes one can use it in conjunction with the Comparison Test. Suppose that the power series ∞ ∞ an (x − x0 )n , bn (x − x0 )n n=0 n=0 have radius of convergence R1 and R2 respectively. If |an | ≤ |bn | for all n, then R1 ≥ R2 . If |an | ≥ |bn | for all n, then R1 ≤ R2 . In short, power series with smaller coefficients have larger radius of convergence. Consider for example the power series expansion for cos x, 1 + 0x − 12 1 x + 0x3 + x4 − · · · = 2! 4! ∞ (−1)k k=0 1 2k x. (2k )! In this case the coefficient an is zero when n is odd, while an = ±1/n! when n is even. In either case, we have |an | ≤ 1/n!. Thus we can compare with the power series ∞ 12 1n 13 14 1 + x + x + x + x + ··· = x 2! 3! 4! n! n=0 which represents ex and has infinite radius of convergence. It follows from the comparison test that the radius of convergence of ∞ (−1)k k=0 1 2k x (2k )! must be at least large as that of the pow...
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