MATH 106A - CALCULUS II
FALL 2005
QUIZ 9
NAME:
Show ALL your work CAREFULLY.
(a) Use the Ratio Test to determine whether the following infiniteseries converges
or diverges.
∞
X
m=1
m!
(2m)!
Consider the limit
lim
n→∞
an+1
an
= lim
n→∞
(n+ 1)!
(2(n+ 1))! ·(2n)!
n!
= lim
n→∞
(n+ 1)!
n!·(2n)!
(2n+ 2)(2n+1)·(2n)!
= lim
n→∞
(n+1)
(2n+ 2)(2n+1)= lim
n→∞
1
2(2n+1)=0<1.
By the Ratio Test, the series converges.
(b) Use the Ratio Test and the Alternating Series Test to determine whether the
following alternating series converges absolutely or conditionally, or diverges. If
it converges, find upper and lower bounds for its limit.
∞
X
n=0
(−1)nn3
2n
We use the Ratio Test to determine if the series converges absolutely.
Consider the limit
lim
n→∞
|an+1|
|an|= lim
n→∞
(n+1)
3
2n+1 ·2n
n3
= lim
n→∞
1
2·n+1
n3
=1
2<1.
By the Ratio Test, the series converges absolutely. (The convergence of
the series (not absolute) can be deduced by using the Alternating Series
Test.) Since the series is alternating, the limit S=P∞
n=0(−1)nn3
2nlies
between any two consecutive partial sums. In particular, Slies between
0=S0and −1
2=S1, i.e., −1
2≤S≤0.
Date: November 28, 2005.
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