Series Challenge Problems
Written By Patrick Newberry
1. Find the sum of the series
1+
1 1 1 1 1 1
1
1
1
1
1
+ + + + + +
+
+
+
+
+ ···
2 3 4 6 8 9 12 16 18 24 27
where the terms are reciprocals of positive integers that are products of only 2’s and 3’s.
2. Find the interval of convergence of
∞
X
n3 xn and find its sum.
n=1
3. Suppose
x3 x6 x9
+
+
+ ···
3!
6!
9!
x4 x7 x10
+
+
+ ···
v =x+
4!
7!
10!
x2 x5 x8
w=
+
+
+ ···
2!
5!
8!
u=1+
Show that u3 + v 3 + w3 − 3uvw = 1.
4. For p > 1, evaluate
1 + 21p + 31p + 41p + · · ·
.
1 − 21p + 31p − 41p + − · · ·
5. Solve the initial value problem
y 00 − 2xy 0 − 2y = 0,
6. Let a0 = a1 = 1,
∞
X
Evaluate
an .
y 0 (0) = 0
y(0) = 1,
n(n − 1)an = (n − 1)(n − 2)an−1 − (n − 3)an−2 .
n=1
7. Show that the series of reciprocals of positive integers that do not have 0 as a digit converges,
and has sum less than 90.
8. Show that the radius of convergence of the power series
∞
X
(pn)!
n=0
(n!)p
xn is
1
pp
for all positive
integers p.
∞
X
(n + p)! n
9. Show that for all positive integers p, q, the power series
x has an infinite radius
n!(n
+
q)!
n=0
of convergence.
10. Using the Maclaurin series for xex , show that
∞
X
n+1
n=0
Z
11. Verify
0
1
∞
X 1
1
dx
=
.
xx
nn
n=1
n!
= 2e.
12. Find the sum of each of the following series:
(a)
(b)
∞
X
n=0
∞
X
xn ,
|x| < 1
nxn−1 ,
|x| < 1
n=1
(c)
∞
X
nxn ,
|x| < 1
n=1
∞
X
n
(d)
2n
n=1
(e)
(f)
(g)
∞
X
n=2
∞
X
n=2
∞
X
n=1
n(n − 1)xn ,
n2 − n
2n
n2
2n
|x| < 1