Math 105, Assignment 6
Due 2015-04-08, 4:00 pm
In the following problems you are expected to justify your answers unless stated otherwise.
Answers without any explanation will be given a mark of zero. The assignment needs to be
in my hand before I leave the lecture room or you will be given a zero on the assignment!
Don’t forget to staple your assignment! You may lose a mark if you do not.
1. Determine whether the following series converge or diverge:
(a)
(b)
(c)
(d)
∞
X
42s π −s
es
s=2015
∞
X
a=5
∞
X
1
a log99 a
sin(2015 log i)
i2015
i=4
Hint: It may be a good idea to examine
∞ X
sin(2015 log i) 2015
i
i=4
∞
X
f =1
(2f )!
f 2015 (f !)2
p
√
n + 4n n + 5 n
√
(e)
3
8n9 − en6 + πn3 − 2015
n=1
Hint: You may assume without proof that for large enough n
p
√
n + 4n n + 5 n
an = √
3
8n9 − en6 + πn3 − 2015
∞
X
is positive and decreasing. Try a limit comparison test with
bn =
1
np
for some p, as done in class.
P
2. Suppose ∞
n=2 an is a geometric series. Suppose that a5 = 112, a7 = 7
(a) Determine all possible values for the ratio, r =
an+1
an
(b) Suppose that r < 0, determine a2 , a9 .
(c) Determine if the series converges and if it does, evaluate it.
(Note: the series starts at n = 2 not n = 0)
3. Determine the center and radius of convergence of the following power series:
1
(a)
(b)
(c)
∞
X
n2
(x + 10)n
n+1
5
n=3
∞
X
n!
(y − 1)n ,
2 nn
n
n=2
Hint: You may need to use the fact that
n
1
e = lim 1 +
.
n→∞
n
∞
X
2 · 4 · 6 · · · (2n − 2)(2n)
n!
n=1
xn
(d) Bonus:
∞
X
fn xn , where fn is the Fibonacci sequence.
n=0
f0 = 1 f1 = 1,
fn = fn−1 + fn−2
n ≥ 2.
You may assume without proof that limn→∞ fn+1
exists.
fn
Fact: One can show that the above power series is the Taylor series of
1
1 − x − x2
4. Suppose we have
√
∞
X
5an+1 n3 − 4 n
n=1
an
n3
+
6n2
−2
=
√
e
π 2015
Determine the convergence of the series:
∞
X
an
n=1
Hint: Use the divergence test on the convergent series. What does that tell you about
the limit
√
5an+1 n3 − 4 n
lim
n→∞ an n3 + 6n2 − 2
5. Find the power series and radius of convergence for the following functions:
(a) sin x, centered at x =
π
3
2
(b) x2015 ex , centered at x = 0.
(c) x2 log(2015 − x), centered at x = 0.
2015
(d)
, centered at x = 1.
(2 + x)2
2