MATH 106A - CALCULUS II
FALL 2005
QUIZ 8
NAME:
Show ALL your work CAREFULLY.
(a) Find the limit of the following sequence or explain why it does not
exist.
an=n(1 −cos(1/n))
Note that as n→∞,cos(1/n)→1so that anapproaches to an
indeterminate form of ∞·0. We need to re-write anand then use
L’Hˆopital’s rule. Write an=(1−cos(1/n))
(1/n). Then
lim
n→∞
an= lim
n→∞
(1 −cos(1/n))
(1/n)
= lim
n→∞
sin(1/n)·(−1/n2)
−(1/n2)by L’Hˆopital’s rule
= lim
n→∞
sin(1/n)
=0.
Thus the sequence {an}converges to 0.
(b) Determine whether the following infinite series converges or diverges.
If it converges, evaluate the infinite series.
∞
X
n=3
3π
4n
This is a geometric series where the leading term a=3π
43with
common ratio r=π
4. Since π
4<1, the geometric series converges
and thus ∞
X
n=3
3π
4n=a
1−r
=3π
43
1−π
4
=3π3
16(4 −π).
Date: November 14, 2005.
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