Student number
Name [SURNAME(S), Givenname(s)]
MATH 101, Section 212 (CSP)
Week 9: Marked Homework Assignment
Due: Thu 2011 Mar 17 14:00
HOMEWORK SUBMITTED LATE WILL NOT BE MARKED
1. Use the Integral Test to determine if the series is convergent or divergent.
(a) 1 +
√1
3
+
√1
5
+
√1
7
+...
1
n=2 n(ln n)3
(b)
P∞
(c)
P∞
n=2
2. Let sn =
ln(n2 )
n
Pn
1
i=1 i
= 1 + 12 + 31 + . . . +
1
n
be the nth partial sum of the harmonic series.
(a) Draw illustrations to show that the partial sums of the harmonic series satisfy
ln(n + 1) =
R n+1
1
1
x
dx ≤ sn ≤ 1 +
Rn
1
1 x
dx = 1 + ln n.
(b) Even though we know the harmonic series is divergent, the partial sums grow (to
∞) very slowly. Suppose you started with s1 = 1 the moment the universe was
formed, and added a new term from the harmonic series every second. Estimate
how large sn would be today, assuming for simplicity that every year consists of
365 days and the universe formed 13 × 109 years ago (use a calculator).
−n
3. Show that ∞
is convergent, and estimate the sum to within 0.1 of its exact
n=1 ne
value (use a calculator).
P
4. How many terms of the convergent series
with error at most 0.00001?
1
n=1 n1.1
P∞
should be used to estimate its sum
5. Determine whether the series is convergent or divergent.
(a)
P∞
(−1)n+1 ln(n2 )
n
(b)
P∞
(−1)n−1 n!
3n
(c)
P∞
(d)
P∞
n=2
n=1
n=1 (−1)
n=1
n
n sin
1
n
sin(nπ/2)
2n
n+1
6. Show that ∞
ne−n is convergent, and estimate the sum to within 0.1 of its
n=1 (−1)
exact value (use a calculator).
P
7. How many terms of the convergent series
sum with error at most 0.00001?
P∞
n=1
(−1)n−1
n1.1
should be used to estimate its