Assignment 6, Math 220
Due: Monday, March 25th, 2002
1 For each of the following series, indicate whether they converge or
diverge and give reasons for your answers.
P
1
a: ∞
k 2 +k
Pk=1
∞
b: P k=1 k −k
1
c: ∞
k log k
Pk=2
d: P ∞
k=1 sin(π/k)
∞
1
e: k=3 k2 −3k+2
P∞ k!
f: k=1 kk
2 For each of the following series, indicate whether they converge or
diverge and give reasons for your answers. Which converge absolutely?
P
(−1)k
√
a: ∞
k=1
k
P∞
k
b: k=1(−1)k k+1
P∞
c: k=1 (−1)k k2k+1
3 −4
P∞
k 2k k 2
d: k=1(−1) k!
P
P 2
3 If
an with an > 0 is convergent, then is
an always convergent?
Either prove it or give a counterexample.
P√
P
an always conver4 If
an with an > 0 is convergent, then is
gent? Either prove it or give a counterexample.
P
5 Can youPgive an example
of
a
convergent
series
xn and a diverP
gent series
yn such that (xn + yn ) is convergent.
P
6 Find an explicit expression for the nth partial sum of ∞
n=2 log(1 −
2
1/n ) to show that this series converges to − log 2. Is this convergence
absolute?
7 Show that the series
1 1 1 1 1 1 1 1
1+ − + + − + + − +···
2 3 4 5 6 7 8 9
is divergent.
1