Math 242: Principles of Analysis
Homework 9
1. Given any real number b (not necessarily rational) and a real number a > 0, we define the
number ab by ab = exp(b ln(a)).
(a) Fix any real number p. Prove that the function f (x) = xp (defined as above) is
differentiable on (0, ∞) and f ′ (x) = pxp−1 .
(b) Fix any real number a > 0 and define g(x) = ax . Prove that g is differentiable on R
and that g ′ (x) = ax ln(a).
2. Use the Fundamental Theorem of Calculus to evaluate the following integrals.
∫ e
1
(a)
dx
1 x
e
∫ 1
(b)
xπ dx
0
∫
1
π x dx
0
∫ π
cos(x) −
(d)
(c)
0
1 dx
2
3. Find the derivative of each function below.
∫ x
2
(a) f (x) =
e−t dt
2
∫
x2
(b) g(x) =
e−t dt
2
∫
3x
ln(1 + t2 ) dt
(c) h(x) =
2x
For the next two problems, recall that the power series expansion
∑
1
=
xn
1 − x n=0
∞
converges for x ∈ (−1, 1).
4. Use substitution and term-by-term integration and differentiation to obtain power series
expansions for each of the following functions.
(a)
1
1 + x4
1
(1 + x4 )2
∫ x
1
(c)
dt
4
0 1+t
(b)
5. (a) Use term-by-term differentiation to find a formula for
∞
∑
n3 xn .
n=1
(b) Find the sum of the series
∞
∑
3
n
and
3n
n=1
∞
∑
n=1
n+1 3
(−1) n
.
3n
(c) Use term-by-term integration to find a formula for
∞
∑
n=1
(d) Find the sum of the series
∞
∑
1
2n n(n
n=1
x
6. (a) Use the power series e =
∫
(b) Express
∞
∑
xn
n=0
2
n!
+ 1)
xn
.
n(n + 1)
.
to write a power series expansion of e−x .
2
e−x dx as a series.
2
0
(c) How many terms of the series in part (b) are needed to approximate the integral to
within 0.0001? (Hint: Corollary 6.2.3)
7. Let f (x) =
x3
.
1 + 2x2
(a) Write a power series expansion for f centered at x0 = 0. On what interval does this
series converge? (Hint: Use the geometric series formula.)
(b) Evaluate f (9) (0).
∫
(c) Write a power series expansion for F (x) =
x
f (t) dt.
0
8. Use the power series expansion sin(x) =
∞
∑
(−1)n x2n+1
n=0
(2n + 1)!
6 sin(x) − 6x + x3
.
x→0
x5
to evaluate lim