HOMEWORK #11
MATH 101 − 205
Question 1 [1 mark]. We write {a n } ≪ {b n } if there exists a number N such that a n ≤ b n for
all n ≥ N (i.e., if all the terms of {b n } are eventually bigger than the corresponding terms in
{a n }). Use this notation to order the following sequences:
{1000n },
{1000n ln(n)},
{10n! },
{10100
1000
},
{102n },
{n n! },
{n1000 },
{ln(n10n )}.
Question 2 [4 marks]. Determine for each of the following series whether they are convergent
or divergent:
∞
∑ ne−n ,
n=1
∞
(n + 1)2
,
2
n=2 n(n − 1)
∑
∞
∑ (−1)n−1
n=100
1
,
ln(ln(n))
∞
sin(2πn)
.
n
n=1
∑
Question 3 [2 marks]. (a) Approximate the convergent series
∞
4
P = ∑(−1)n
2n + 1
n=0
to within an error of 0.2.
(b) Approximate the convergent series
∞
(−1)n (6n)!(13591409 + 545140134n)
(3n)!(n!)3 6403203n+3/2
n=0
Q = 12 ∑
to within an error of 10−15 . Then use this to approximate Q −1 (no error bounds required here).
Obvious hint: use a calculator. Less obvious hint: use that 0! = 1.
Question 4 [3 marks]. The Sierpinski triangle is a fractal that can be constructed as follows:
Start with an equilateral triangle with area 1. Remove from its interior the triangle whose
corners are located at the midpoints of the sides. This results in configuration of three touching
equilateral triangles. For each of these triangles, remove again the interior triangles. Repeat
the process infinitely often.
Figure 1. The first five steps of the construction of the Sierpinski triangle.
Express the area removed from the triangle by the construction of the Sierpinski triangle as
a series and show that this series converges to 1.
1Due:
March 26, 2014 (at the beginning of class). Instructor: Dr. T. Hulshof.
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