Math 222 Calculus and Analytic Geometry
Wisconsin Emerging Scholars
Sep. 26th
Worksheet 8
Infinite Series
Problem 1. Decide whether the following series converge or not. If it does, then find
the value of infinite series.
A. 2 − 1 +
1 1 1
1
− + −
+ ···
2 4 8 16
B. 1 − 1 + 1 − 1 + 1 − 1 + · · ·
C.
D.
1
1
1
1
+
+
+ ··· +
+ ···
2·3 3·4 4·5
(n + 1) · (n + 2)
∞
X
22n
n=1
E.
3n
∞
X
(−1)n
n=0
4n
1
∞ X
5
1
F.
− n
n
2
3
n=1
G.
∞
X
4n − 5n+1
n=0
H.
32n
∞
X
cos nπ
n=0
5n
2
Problem 2. Calculate the following series.
∞ X
1
1
√ −√
A.
n
n+1
n=1
B.
∞
X
n=1
6
(2n − 1)(2n + 1)
∞ X
√
√ C.
ln n + 1 − ln n
n=1
D.
∞
X
tan−1 (n) − tan−1 (n + 1)
n=1
3
Problem 3. Find the values of x for which the given geometric series converges. Also,
find the sum of the series (as a function of x) for those values of x.
A.
∞
X
(−1)n x−n
n=0
B.
∞
X
sinn x
n=0
Problem 4. True or False?
A.
∞
X
e−n converges.
n=0
B. If
∞
X
an and
n=0
C. If
∞
X
n=0
D. If
∞
X
n=0
∞
X
bn converge, then so does
n=0
an and
∞
X
∞
X
(an + bn ).
n=0
bn diverge, then so does
n=0
an converges and
∞
X
(an + bn ).
n=0
∞
X
bn diverges, then
n=0
∞
X
n=0
4
an bn diverges.