Infinite Series
Math 236 Spring 2001
This worksheet is a review of the types and techniques of analyzing infinite series. You do not have to hand
in this worksheet. At least two of these problems will be on the next exam.
1.
Express
1
1
1
1
1
+
+
+
+ ... +
in sigma notation.
12 20 30 42
90
2.
Express
33
32
1
+
+ . . . + 2 7 in sigma notation:
2
3
e
e
3 e
(a)
Starting from k = 3.
(b)
Starting from k = 2.
(c)
Starting from k = 0.
3.
Show that the nth partial sum for the geometric series
∞
X
xk is sn =
k=0
4.
Find a formula for the nth partial sum sn for the series
∞
X
k=1
5.
1 − xn+1
.
1−x
1
.
k(k + 2)
Determine whether the following series converge or diverge. If convergent, find the sum.
(a)
∞
X
3
ek
k=1
∞
X
2
(2k − 1)(2k + 1)
(c)
1−
2
4
8
+
−
+ ...
5 25 125
(d)
∞
X
(b)
k=1
k=1
6.
7.
k2
1
+ 3k + 2
Consider the rational number α = 0.251251251251251 . . .
(a)
Write the number α as an infinite series.
(b)
Find the sum of the series and use it to express α as the quotient of two integers.
Find a series expansion for
x
, |x| < 1.
1 + x3
8.
Determine whether the following series converge or diverge. Justify your answers.
(a)
∞
X
2k
1 + k4
k=1
(b)
∞ X
k=1
(c)
(e)
1
k(ln k)3
(m)
(n)
∞
X
(o)
1
2 + 3−k
∞
X
k=1
(h)
1
√
k k
∞
X
k!
104k
∞
X
1
cosh2 k
k=1
k!
∞
X
1
√
2−1
k
k
k=1
∞
X
∞
X
k=1
(p)
1
3k − 2
∞
X
ln k
k=1
k
√
k
2k 2 + 1
k
∞ X
3k
2k + 1
k=1
(q)
k=1
(i)
∞
X
10k
k=0
∞
X
k2
ek
k2
2k + 1
k=3
∞
X
k=6
(g)
(k)
(l)
k=0
(f)
k
1
(k + 3)(k + 4)
k=2
∞
X
k=1
∞
X
k=1
(d)
k
2k + 100
(j)
∞
X
32k
(2k)!
k=0
(r)
∞
X
ln k
k=1
kk
9.
Determine whether each series converges absolutely, converges conditionally, or diverges.
(a)
∞
X
(−1)k
√
k
k=1
(d)
∞
X
(−1)k
√
k+ k
k=1
(b)
∞
X
(−1)k k
(e)
∞
X
(−1)k+1 2k
∞
X
(−1)k+1
(f)
k=1
(c)
k=1
10.
k+2
3k + 4
k=1
3k + 1
∞
X
(−1)k+1 2k
k2
k=5
Find the smallest integer n so that sn will approximate
∞
X
(−1)k
to within 0.01.
k2 + 2
k=0