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In more advanced courses, one studies criteria under which functions are
real analytic. For our purposes, it is suﬃcient to be aware of the following facts:
The sum and product of real analytic functions is real analytic. It follows from
this that any polynomial
P (x) = a0 + a1 x + a2 x2 + · · · + an xn
is analytic at any x0 . The quotient of two polynomials with no common factors,
P (x)/Q(x), is analytic at x0 if and only if x0 is not a zero of the denominator
Q(x). Thus for example, 1/(x − 1) is analytic whenever x0 = 1, but fails to be
analytic at x0 = 1.
1.1.1. Use the ratio test to ﬁnd the radius of convergence of the following power
(−1)n xn ,
∞ ∞ 3
(x − 2)n ,
n=0 c. d. ∞ 1
(x − π )n ,
∞ (7x − 14)n , e. f. n=0 1
(3x − 6)n .
n=0 1.1.2. Use the comparison test to ﬁnd an estimate for the radius of convergence
of each of the following power series:
k=0 ∞ 1 2k
(2k )! (−1)k x2k , b.
(x − 4)2k
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This document was uploaded on 01/12/2014.
- Winter '14