Analysis Homework #8
due Thursday, Mar. 31
1. Test each of the following series for convergence:
(
)
∞
∑
1
1
· log 1 +
,
n
n
n=1
∞
∑
(−1)n e1/n
n=1
n
.
2. Find the radius of convergence for each of the following power series:
∞
∑
nxn
n=0
3n
,
∞
∑
(n!)2 n
·x .
(2n)!
n=0
3. Assuming that |x| < 1, use the formula for a geometric series to show that
∞
∑
n=0
nxn =
x
.
(1 − x)2
4. Let α ∈ R be fixed. Find the radius of convergence for the binomial series
f (x) =
∞
∑
α(α − 1)(α − 2) · · · (α − n + 1)
n=0
n!
· xn .
• You are going to work on these problems during your Thursday tutorial (at 4 pm).
• When writing up solutions, write legibly and coherently. Use words, not just symbols.
• Write your name and then MATHS/TP/TSM on the first page of your homework.
• Your solutions may use any of the results stated in class (but nothing else).
• NO LATE HOMEWORK WILL BE ACCEPTED.