Math 3210-1
HW 17
Due Tuesday, July 31
Convergence of Infinite Series
1. Find the sum of each series:
∞ n
X
1
(a)
2
n=3
n
∞
X
2
−
(b)
3
n=1
(c)
∞
X
1
n(n + 1)(n + 2)
n=1
∞
X
1
√
√ converges. Justify your answer.
n+1+ n
n=1
P
P
3. Prove that if
|an | converges and (bn ) is a bounded sequence, then
an bn converges.
2. Determine whether or not the series
Convergence Tests
4. Determine the values of p for which the series
∞
X
1
converges.
n(log
n)p
n=2
5. Determine whether each series converges conditionally, converges absolutely, or diverges. Justify your
answers.
(a)
∞
X
(−1)n
log n
n=1
(b)
X (−2)n
n2
X (−3)n
(c)
n!
X 1
1
√ −
(d)
n n
6. Find an example to show that
P the convergence of
imply the convergence of (an bn ).
P
an and the convergence of
P
bn do not necessarily
7. Show that the series
1 1
1
1
1
1
1
+ − 2 + − 3 + − 4 + ···
2 3 2
5 2
7 2
diverges. Why doesn’t this contradict the alternating series test?
1−
Power Series
8. Find the radius of convergence R and the interval of convergence C for each series:
(a)
X n2
2n
xn
(b)
(c)
X (−4)−n
n
X
xn
(2−n )(x − 5)2n
9. Find the radius of convergence for
X (3n)!
(n!)2
xn .
P
10. Suppose that the series
an xn has radius of convergence 2. Find the radius of convergence of each
series, where k is a fixed positive integer.
X
(a)
akn xn
X
(b)
an xkn
X
2
(c)
an xn
11. Prove that the series
∞
X
n=0
an xn and
∞
X
n=0
nan xn have the same radius of convergence (finite or infinite).