Math 121: Problem set 11 (due 4/4/12)
Practice problems (not for submission!)
Sections 9.5-9.7
Sequences and Series
1. Find the center, radius, and interval of convergence for the following series:
10en +5 n
(a) ∑∞
n=0 √ nn x
n
n+5
(b) ∑∞
n=0 4n (3x − 1)
2. Evaluate the following sums:
(−1)n+1
(a) ∑∞
n=2 n3n .
1
(b) ∑∞
n=0 n2 2n (express your answer as an integral).
23
13
33
(*c) x3 − 3!x · 9 + 5!x· 81 − 7!x· 729 + · · ·
2
4
1
2n
(*d) 1 + x3! + x5! + · · · = ∑∞
n=0 (2n+1)! x .
2
n
PRAC ∑∞
n=0 (n − 3n + 5)x where |x| < 1.
3. Expand
(a) x13 about x0 = 2 (Hint)
x
(b) 1+3x
2 in powers of x.
(c) 11+x logt
t−1 dt in powers of x.
PRAC x3 +x21−5x+3 in powers of x.
R
4. (Taylor polynomials)
(a) Evaluate limx→0
(b) Find r so that
sin(sin x)−x cos( 37 x)
arctan x(ex −1−x)2
sin x −ex
A = limx→0 xr eecos x−1
exists and is non-zero, and evaluate A.
2n−1
5. Define a sequence by a0 = a1 = 1 and an+2 = (n+1)(n+2)
an if n ≥ 0.
∞
n
(a) Show that the series ∑n=0 an x converges for all x.
(b) Let f (x) denote the sum of the series in part (a). Show that f 00 (x) − 2x f 0 (x) + f = 0.
131
Hint for 2(c): Differentiate the unknown function.
Hint for 3(a): Expand 1x first, then differentiate.
Hint for 3(d): Partial fractions.
132