Math 166Z Homework # 7.5
Ungraded
1. For each of the following series determine whether it converges or diverges. If converges,
find its sum.
n−1
∞
X
3
(a)
2
4
n=1
∞
X
1
(b)
e2n
n=1
(c)
∞
X
n=1
1
n(n + 2)
2. For each of the following series determine whether it is convergent of divergent.
∞
X
1
(a)
n ln n
n=2
(b)
(c)
(d)
∞
X
ln n
n=1
∞
X
k=1
∞
X
k=1
n
k−1
2k + 1
k2
e−k
3. Find the following limits:
ln x
(a) lim
x→1 x − 1
(ln x)3
(b) lim
x→∞
x2
√
(c) lim+ x sec x
x→0
1
(d) lim (xe x − x)
x→∞
(e) lim+ xsin x
x→0
3
3
(f) lim
−
x→0
x sin x
4. Evaluate each of the following integrals or state that it is divergent.
Z 1
x ln x dx
(a)
0
Z ∞
x
(b)
dx
2
−∞ 1 + x
Z ∞
dx
(c)
x(ln x)2
e
5. On the last homework you wrote repeating decimals as geometric series. For example,
3
+ 1032 + 1033 + 1034 + . . . The denominators of these fractions are 10 because
.33333 . . . as 10
we usually use a base 10 (decimal) number system. However, we could also choose to
use other bases (the babylonians used base 60, which is why there are 60 seconds in a
minute and 60 minutes in an hour.). Base 2 (binary) is probably the next most common
base used after base 10.
The following are written in base 3 (Ternary). Write each of them as a geometric series
and find its sum then use this result to write the fraction ab where a and b are integers
(you don’t have to write the integers in Ternary).
(a) 0.12 = 0.12121212...
(b) 0.2 = 0.222222...
(c) 0.1112 = 0.1112121212...
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Here is a picture of the babylonian numerals. It doesn’t have anything to do with the
homework, so you don’t have to print it off.
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