Math 257 – Assignment 1
Due: Wednesday, January 12
1. Compare the following series to an integral and show that:
a)
∞
!
ne
−n2
converges
b)
n=1
∞
!
n=2
1
diverges
n ln(n)
2. Determine whether the following series are convergent or not:
a)
c)
∞
!
n2
n=0
∞
!
n=1
3. Show that:
a)
n=3
(−1)n
2n + 1
∞
!
(−1)n
n=0
b)
2n
2n
∞
!
n+4
2
=
3
b)
n2
∞
!
1
1
=
n−1
3
2
n=2
4. Find the radii of convergence of the following power series:
a)
c)
∞
!
xn
(2n)!
n=1
∞
!
b)
∞
!
n2 xn
n=0
2n xn
n=0
5. Find the Taylor series of f (x) = ln(1 + x) around x = 0. Determine the radius of convergence of the series and discuss its convergence behaviour at the end points of the convergence
interval.
6. "
Consider the equation y # + y = 0. Find a power series solution of the form y(x) =
∞
n
n=0 an x . Then solve the equation directly to confirm your series solution.