Math 166Z Homework # 10
Due Thursday April 21th
1. Find the radius of convergence and the interval of convergence for the following series:
∞
X
(3x)n
(a)
n+2
n=0
(b)
(c)
∞
X
n=0
∞
X
n=1
n xn
(−1)n x2n−1
(2n − 1)!
2. Find the power series representation for the following functions and determine the interval of convergence:
x
(a) f (x) =
1−x
1
(b) f (x) =
4 + x2
(c) f (x) = arctan(2x)
(d) f (x) = xex
2
(e) f (x) = ex + e−x
3. Find the sum of the following series by recognizing how it is related to something familiar:
cos2 x cos3 x cos4 x
(a) 1 + cos x +
+
+
+ ...
2!
3!
4!
(b) x − x2 + x3 − x4 + x5 − . . .
4. Find the Taylor series centered at a = 1 for the function f (x) =
(i.e f (x) = f (a) + f 0 (a)(x − a) + . . . ).
1
using the definition
x
5. Find the Taylor series through the (x − a)5 term for the following functions centered at
3
the given value of a. It might be easier to use known series (such as sin x = x − x3! + . . . )
and then perform multiplications, divisions, etc.:
(a) f (x) = sin(2x), a =
π
4
(b) f (x) = sec(x), a = 0
(c) f (x) = cos2 (x), a = π
Z
6. Use series to approximate the definite integral
0
1
sin(x2 ) dx to three decimal places.