MATH 141, FALL 2016, HW12
DUE IN SECTION, DECEMBER 1
Problem 1. A sequence {an }∞
n=1 is defined recursively in terms of a1 and a2
by
an + an−1
,
n ≥ 2.
an+1 =
2
Prove that for any real a1 and a2 , {an }∞
n=1 converges, and compute the limit.
Problem 2. A function f satisfies the differential equation
xf 00 (x) + 3x[f 0 (x)]2 = 1 − e−x
for all real x.
(1) If f has an extremum at a point c 6= 0, show that this extremum is a
minimum.
(2) If f has an extremum at 0, is it a maximum or a minimum? Justify
your conclusion.
(3) If f (0) = f 0 (0) = 0, find the smallest constant A such that f (x) ≤ Ax2
for all x ≥ 0. (Hint: 1 − e−x < x for x > 0.)
Problem 3. Decide convergence of the following improper integrals.
R1
√ x dx
(1) 0+ log
x
R 1− log x
(2) 0 1−x dx
R 1−
(3) 0+ √xdx
R ∞ dxlog x
(4) 2 x(log x)3 .
Problem 4. For a certain real C, the integral
Z ∞
x
C
−
dx
2 + 2C
2x
x
+
1
1
converges. Determine C and evaluate the integral.
1
2
DUE IN SECTION, DECEMBER 1
Problem 5. Find the radius of convergence of each of the following power
series. Within the radius of convergence, evaluate the series in terms of
elementary functions.
P
3 n
(1) P∞
n=0 nn z
2 n
(2) ∞
n=0 n2 z
P∞
n 2
(3) n=0 2 n!n z n
P
n3 n
(4) ∞
n=0 3n z .
P
Problem 6. Let an ≥ 0. Prove that the convergence of ∞
n=1 an implies the
convergence of
∞ √
X
an
.
n
n=1
Bonus Problem. Prove that
f (x) =
X 2k k≥0
k
xk
has radius of convergence 14 , and, within its radius of convergence,
1
.
f (x) = √
1 − 4x