MATH 106A - CALCULUS II
WINTER 2007
QUIZ 8
NAME:
Show ALL your work CAREFULLY.
(a) Use the Integral Test to determine whether the following infinite series con-
verges or diverges.
∞
X
j=2
ln j
j2
Compare the series with the improper integral R∞
2
ln x
x2dx. Note that
the for x≥2, the function ln x
x2is decreasing. Now,
Z∞
2
ln x
x2dx = lim
b→∞ Zb
2
ln x
x2dx
IBP
= lim
b→∞
(ln x)·(−x−1)
b
2
−Zb
2
(−x−1)·1
xdx
= lim
b→∞
−ln b
b+ln 2
2−1
b+1
2
=ln 2 + 1
2<∞.
Thus, the improper integral converges and by the Integral Test so does
the infinite series.
(b) Use the Ratio Test to determine whether the following infinite series con-
verges or diverges.
∞
X
n=1
n2
2n
Here, an=n2
2n. We have
lim
n→∞
an+1
an
= lim
n→∞
(n+1)
2
2n+1 ·2n
n2
= lim
n→∞ n+1
n2
·1
2=1
2<1.
By the Ratio Test, the series converges.
Date: March 23, 2007.
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