Math 240
Spring 2006
Dr. Khalili
MAPLE PROJECT 3
Due date April.14, 2006
Problem 1. The harmonic series
∞
X
n=1
1
n= 1 + 1
2+1
3+1
4+· · · +1
n+· · ·
is one of the most impotant series in chapter 8 (infinite series.) Note that the general term of the series, 1
n,
goes to zero as n goes to infinity, however the series diverges!
In other words, even though the terms are getting smaller and smaller, the sum “adds up to infinity.”
(a) One way to show that the harmonic series diverges is due to J. Bernoulli. He grouped the terms of
the harmonic series as follows:
1 + 1
2+1
3+1
4
| {z }
>1
2
+1
5+· · · +1
8
| {z }
>1
2
+1
9+· · · +1
16
| {z }
>1
2
+1
17 +· · · +1
32
| {z }
>1
2
+· · ·
Write a short paragraph explaining how you can use this grouping to show that the harmonic series
diverges.
(b) How many terms Myou need so that
M
X
n=1
1
n>50
(c) Show that the sum of the first million terms of the harmonic series is less than 15.
(d) Show that the following inequalities are valid.
ln21
10 ≤1
10 +1
11 +1
12 +· · · +1
19 +1
20 ≤ln20
9
ln201
100 ≤1
100 +1
101 +1
102 +· · · +1
199 +1
200 ≤ln200
99
(e) Find the following limit in two different ways (i) using the ideas in part (d) and (ii) using the limit
command of MAPLE.
lim
m→∞
2m
X
n=m
1
n