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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
x + 0x3 + x4 − · · · =
4! ∞ (−1)k
k=0 1 2k
(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
1 + x + x + x + x + ··· =
which represents ex and has inﬁnite radius of convergence. It follows from the
comparison test that the radius of convergence of
k=0 1 2k
(2k )! must be at least large as that of the pow...
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- Winter '14