Math 105/206 - Quiz 6, Apr 10 2015
IMPORTANT: Write your name AND student number somewhere on this sheet.
No calculators, books or notes. Please show your work to get full marks. (10 + 2 marks total)
Problem 1
For each of the following series, say if it converges or diverges and motivate your answer (3 marks each).
∞
X
k=37
∞
X
k=1
k2
1
+ 2k + 1
(−1)k
ek
1
+1
Problem 2
Determine the radius of convergence of the following power series centred in 0. (3 marks)
∞
X
(−10)k
k=1
k 3k
· xk
Problem 3
Find the Maclaurin series (=Taylor series centred in a = 0) of the function
f (x) =
x
− arctan(x).
1 + 2x2
What is its interval of convergence? (3 marks)
Hint: recall that
∞
X
1
=
xk
1 − x k=0
and
arctan(x) =
∞
X
k=0
for |x| < 1.
(−1)k
x2k+1
2k + 1