Homework 2 Solutions
Problem 2.1 (practice): Series notation I Write out the first four nonzero
terms in the series:
a)
∞
X
1
n!
n=0
Solution:
∞
X
1
n!
n=0
=
1
1
1
1
+ + + + ...
0! 1! 2! 3!
=
1+1+
1 1
+ + ...
2 6
b)
∞
X
(−1)n
n!
n=1
Solution:
∞
X
(−1)n
n!
n=1
=
−11
−12
−13
−14
+
+
+
+ ...
1!
2!
3!
4!
= −1 +
1
1 1
− +
+ ...
2 6 24
Problem 2.2P(practice): Series notation II
using sigma ( ) notation.
Write the following series
a)
1 − 2 θ2 + 4 θ4 − 8 θ6 + . . .
Solution:
∞
X
−2n θ2n
n=0
b)
1 1
1
1
− +
−
+ ...
4 9 16 25
1
Solution:
∞
X
−1n
n2
n=2
Problem 2.3
P (practice): Series notation III If you need more practice
with sigma ( ) notation, you can get really good practice by going back and
forth between the two representations of the standard power series on the memorization page. Power series are used everywhere in physics and it is very important to be able to translate back and forth between the two representations.
Problem 2.4: Tetrahedron Find the angle between any two line segments
that join the center of a tetrahedron to its vertices. Hint: Think of the vertices
of the tetrahedron as sitting at the vertices of a cube. (You may need to build
a model and play with it to see how this works!)
Solution: Think of one of the vertices of the tetrahedron lying at the origin,
and the other vertices at (1, 1, 0), (1, 0, 1), and (0, 1, 1). The center of the tetrahedron is at ( 12 , 12 , 12 ). Consider two vectors each pointing from the center of the
tetrahedron to one of the vertices (say the first two on the list above). Then we
can calculate the dot product of these two vectors in two ways: algebraically
and geometrically.
1
1
1
1
1
1
(ı̂ + ̂) − ( ı̂ + ̂ + k̂)
·
(ı̂ + k̂) − ( ı̂ + ̂ + k̂)
2
2
2
2
2
2
1
1
1 1
1
1 = ı̂ + ̂ − k̂ ı̂ − ̂ + k̂ cos γ
2
2
2
2
2
2
r r
1
1
1
1
1
1
3 3
ı̂ + ̂ − k̂ ·
ı̂ − ̂ + k̂
=
cos γ
2
2
2
2
2
2
4 4
1
3
=
cos γ
−
4
4
1
cos γ = −
3
1
γ = cos−1 −
3
= ≈ 109o
Problem 2.5: Tangent Power Series Find the first four terms in the power
series of tan θ. Do so by using the set of power series that you are asked to
memorize.
2
Solution:
We begin with
1 3
1
x + x5 + · · ·
3!
5!
1
1
cos x = 1 − x2 + x4 + · · ·
2
4!
1
= 1 − x + x2 − x3 + · · ·
1+x
sin x = x −
(1)
(2)
(3)
The third line above can come either from the (1 + z)p power series (with
p = −1), or by taking a derivative of the ln(1 + z) power series. The latter is
actually a little easier.
Given these, we can write
sin θ
cos θ
1 3
1 5
θ − 3!
θ + 5!
θ + ···
=
1 2
1 4
1 − 2 θ + 4! θ + · · ·
tan θ =
(4)
(5)
1
Now the bottom looks a lot like 1+θ
, so we can use that power series:
1
1
1
1
1
1
1 − (− θ2 + θ4 + · · · ) + (− θ2 + θ4 + · · · )2 + · · ·
tan θ = θ − θ3 + θ5 + · · ·
3!
5!
2
4!
2
4!
(6)
We now just need to multiply this all out and collect terms with the same power,
making sure that we have all the terms with that power, and drop the higher
order terms. Since we only want the first three terms (up to θ5 ), we can start
dropping out higher terms than that.
1 3
1 5
1 2
1 4
1 2
1 4
2
tan θ = θ − θ + θ + · · ·
1 − (− θ + θ + · · · ) + (− θ + θ + · · · ) + · · ·
3!
5!
2
4!
2
4!
(7)
1
1
1
1
1
= θ − θ3 + θ5 + · · ·
1 + θ2 − θ4 + θ4 + · · ·
(8)
3!
5!
2
4!
4
1
1
1
5
= θ − θ3 + θ5 + · · ·
1 + θ2 + θ4 + · · ·
(9)
3!
5!
2
4!
1
5
1
1
1
= (θ + θ3 + θ5 + · · · ) − (θ3 + θ5 + · · · ) + (θ5 + · · · ) + · · · (10)
2
4!
3!
2
5!
2
16
= θ + θ3 + θ5 + · · ·
(11)
3!
5!
1
2
= θ + θ3 + θ5 + · · ·
(12)
3
15
You could have found this power series by taking five derivatives of tan θ (if
the question hadn’t stated otherwise), but that would have been rather more
painful.
3
Problem 2.6: Series convergence Recall that, if you take an infinite number of terms, the series for sin z and the function itself f (z) = sin z are equivalent
representations of the same thing for all real numbers z, (in fact, for all complex
numbers z). This is not always true. More commonly, a series is only a valid,
equivalent representation of a function for some more restricted values of z. The
technical name for this idea is convergence–the series only “converges” to the
value of the function on some restricted domain.
1
Find the power series for the function f (z) = 1+z
2 . Then, using the Mathematica worksheet from class:
http://physics.oregonstate.edu/~roundyd/COURSES/ph320/schedule/
week1/thursday/02-Approximations-using-power-series/03-Power-series-with-Mathematica.
nb 1 as a model, explore the convergence of this series. Where does your series for this new function converge? Can you see the reflection of the region of
convergence in the graphs of the various approximations? Print out a plot and
write a brief description (a sentence or two) of the region of convergence.
Solution: See the Mathematica worksheet http://physics.oregonstate.
edu/~roundyd/COURSES/ph320/schedule/week1/thursday/02-Approximations-using-power-series/
03-Power-series-with-Mathematica.nb which can be found as a link on the
class syllabus.
Problem 2.7: Euler’s formula
a) Work out the power series for eix , where i2 = −1.
b) What is the real part of this power series?
c) What is the imaginary part of this power series?
d) How is eix related to sin x and cos x?
Solution:
a) The power series for eix is
eix = 1 + ix −
i
1
i
1 2
x − x3 + x4 − x5 − · · ·
2!
3!
4!
5!
b) The real part is just
eix = 1 −
1 2
1
x + x4 − · · ·
2!
4!
c) The imaginary part is just
eix = x −
1 the
1 3
1
x − x5 + · · ·
3!
5!
link is called “Power series with Mathematica”
4
d) A quick glance at the power series for cos x and sin x shows that
eix = cos x + i sin x
5