MATH 101 V2A
March 20th – Practice problems
Hints and Solutions
Practice Problems:
1. Find a power series which converges on the interval [2, 8).
Solution: The series
n
∞
X
1 x−5
n
3
n=1
is one example.
2. Find a series which converges to e. You may assume that there is a power series
∞
X
an xn which
n=0
converges to ex for all x < R (for some R > 1), but your answer cannot contain the coefficients an (i.e.
you must find an expression for an in terms of n).
Note: This problem was given before we covered Taylor series, so the solution does not
use any Taylor series.
Solution: Since
d x
dx e
= ex , we get that
∞
X
an nxn−1 =
n=1
X
an xn ,
n=0
and so we can conclude that (n + 1)an+1 = an for all n ≥ 0. We also know that
Z x
et dt = ex − 1,
0
so
∞
X
n=0
an
∞
X
xn+1
= −1 +
an xn .
n+1
n=0
In particular, this means that a0 − 1 = 0. So, a1 =
1
in general, an = n!
.
1
1
· a0 = 1, a2 =
1
2
· a1 =
1
2! ,
a3 =
3. Determine the radius and interval of convergence for each of the following series.
(a)
∞
X
(−1)n
.
n5 5 n
n=1
(b)
∞
X
(1 + 3n )xn
.
n!
n=0
Solution: See the solutions to the practice problems from March 18th.
1
3
· a2 =
1
3!
and,