PHZ 3113 Fall 2010
Homework #1, Due Friday, September 3
α n
. What is lim an ? What is lim an ?
1. Let an = 1 +
n→∞
n
n→0
2. Decide whether the following sums converge or diverge. Present the reasons for your
choices.
∞
∞
∞ X
X
X
1
1
sin n
1
(a)
(b)
(c)
−
n ln n ln(ln n)
n
2n + 1 2n
n=2
n=1
n=1
3. Consider the following alternating series
2
1
2
1
2
1
2
1
2
1− √ + √ − √ + √ − √ + √ − √ + √ − √ +··· .
2
3
4
5
6
7
8
9
10
Does the series converge?
4. A quantum rotator with moment of inertia I has energy levels ` = `(` + 1) h̄2 /2I, with
degeneracy 2`+1, where ` is an integer and h̄ is the (reduced) Planck constant. The partition
function for a system with these energy levels is
Z=
∞
X
(2` + 1)e−β` ,
`=0
where β = 1/kT .
(a) Determine for what values of β the series converges.
(b) Compute an approximate value for Z when T is large. Compute an approximate value
for Z when T is small.
P
5. Take the alternating harmonic series (−1)n+1 /n and sum it in the following order:
take three odd (positive) terms, followed by the first even (negative) term, followed by three
more odd terms, one more even term, three odd terms, one even term, etc.
s=1+
1 1 1 1 1
1
1
1
1
1
1
+ − + + +
− +
+
+
− +··· .
3 5 2 7 9 11 4 13 15 17 6
Does this series converge? If so, what is the value of the sum? Since infinite precision, even
if achievable, is not practical, make some choices and include an estimate of the accuracy of
your result.