Math 3210-3
HW 25
Due Friday, November 30, 2007
Convergence Tests
1. Determine the values of p for which the series
∞
X
1
converges.
n(log n)p
n=2
2. 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
(c)
n2
X (−3)n
n!
X 1
1
√ −
(d)
n n
3. Find an example to show that
P the convergence of
imply the convergence of (an bn ).
4. Show that the series
P
an and the convergence of
1
1
1
1
1
1 1
+ − 3 + − 4 + ···
+ −
2 3 22
5 2
7 2
diverges. Why doesn’t this contradict the alternating series test?
1−
P
bn do not necessarily