Problems for Advanced Calculus Math 522
(1) Determine whether the innite series
k=1
3k 1 2k
,
6k
k=1
k
k2 + 1
converge. If they converge nd their sum.
(2) Test the series for convergence
k+1 k
1
,
.
(ln k )k
k
k=2
k=1
(3) Determine whether the series conv
Advanced Calculus Math 522
Homework 9
1. Evaluate the double integral R (2 x2 y 2 ), where R = [0, 1] [0, 1].
Which theorems do you use?
Solution: The function f (x, y ) = 2 x2 y 2 is continuous on R, so it
is Riemann integrable by Theorem 11.1.5. By Theo
Advanced Calculus Math 522
Homework 8
1. Polar coordinates are given by x = r cos , y = r sin . Use the chain rule
to show that
1
1
fxx + fyy = grr + 2 g + gr ,
r
r
where f (x, y ) and g (r, ) are connected by g (r, ) = f (r cos , r sin ).
Solution: We co
Advanced Calculus Math 522
Homework 7
1. Show that the function
xy (x2 y 2 )
x2 + y 2
if (x, y ) = (0, 0)
0
f (x, y ) =
if (x, y ) = (0, 0)
is dierentiable at every point (a, b) R2 .
Solution: For (x, y ) = (0, 0) we calculate
fx =
fy =
y (x4 y 4 + 4 x2 y
Advanced Calculus Math 522
Homework 6
x2 + y 3
exists.
(x,y )(0,0) x2 + y 2
x2 +y 3
Solution: Let f (x, y ) = x2 +y2 . If x = 0 then f (0, y ) = y so limy0 f (0, y ) =
0. If y = 0 then f (x, 0) = 1 so limx0 f (x, 0) = 1. We found two dierent
limits along
Advanced Calculus Math 522
Homework 5
sin x
dx as an
x
0
innite series. Use this series to evaluate the integral to four decimal places.
Solution: We know that, for all x R,
1. Use the Taylor series for sin x to express the integral
(1)k
sin x =
k=0
x2k+1
Advanced Calculus Math 522
Homework 4
1
, x R. Prove that F is continuously dieren+ k2
k=1
tiable and express its derivative as an innite series.
Solution: We apply Theorem 8.4.17. The term-by-term dierentiated series
is
2x
.
(x2 + k 2 )2
1. Let F (x) =
x
Advanced Calculus Math 522
Homework 3
1. Let fn (x) = nxenx , 0 x 1.
(a) Find the pointwise limit f (x) = limn fn (x).
(b) Does cfw_fn converge uniformly to f on [0, 1]?
Solution: (a) The pointwise limit is f (x) = limn fn (x) = 0 as we see
from Example
Advanced Calculus Math 522
Homework 2
k
k
(x
1. Find all values of x for which the series (1)k+13) (a) converges
k=0
absolutely (b) converges conditionally (c) converges (d) diverges.
Solution: (a) We consider
k=0
(1)k (x 3)k
=
k+1
k=0
|x 3|k
.
k+1
Since
Advanced Calculus Math 522
Homework 1
1. Find the sum of the innite series
k=1
(2)k + 3k
.
4k
Solution:
k=1
(2)k + 3k
=
4k
k=1
k
1
2
3
4
+
k=1
k
=
3
1
2
+4
1+ 1
1
2
3
4
1
8
= +3 = .
3
3
1
for convergence or divergence.
k (ln k )2
k=2
Solution: We are usin
Problems for Advanced Calculus Math 522
10.1 For the set S = cfw_(x, y ) : y > x nd int S , S , S , S . Is the set S
open, closed, bounded or arcwise connected?
10.2 Determine the limit (if it exists)
sin(x2 + y 2 )
.
x2 + y 2
(x,y )(0,0)
lim
10.3 Let f (
Advanced Calculus Math 522
Homework 10
1. Show that if f is continuous on R, then
x
y
x
(x y )f (y ) dy.
f (x) dx dy =
0
0
0
Solution: We interchange order of integration
x
y
x
x
f (t) dt dy =
0
0
x
(x t)f (t) dt.
f (t) dy dt =
0
t
0
2. Determine a value