Homework 10
due on Monday, 11 May 2009 by 12:50 PM
1. Assume that the series
n=1
an 3n converges conditionally.
(a) What can you say about the convergence of
(1)n an 3n ? Explain your reasoning.
n=1
(b) Determine whether the series (c) Is the series an n+
Solutions of Homework 9 Question 1. Let g be a decreasing dierentiable function on (100, ). Assume that
x
lim x3 g (x) = 0
and
x
lim g (x) = 1.
Show that the series
1 g (n) is convergent.
Solution: As g is decreasing and as lim g (x) = 1, it follows that
Due on April 20, up to 12:50 p.m. Math112 Homework 9
Question 1. Let g be a decreasing dierentiable function on (100, ). Assume that
x
lim x3 g (x) = 0
and
x
lim g (x) = 1.
Show that the series
1 g (n) is convergent. (1)n sin 1 np absolutely convergent,
Q
Math.112 ,HW 8 Due on Apr. 13,2009 up to 12:50 p.m.
1. Find the sum of the series
(n
n =2
2
2n + 1 . 1) ( n + 2 ) n
. ln n (Hint: Remember the proof of the Integral Test) 2. Evaluate lim
k =1 n
k
n
1
Determine whether the following series is convergent
Math.112 ,HW7 Due on Apr. 6,2009 up to 12:50 p.m. In problems 1,2, and 3,find the limit of the sequence cfw_a n if the limit exits
1 1 1 1. a n = 1 2 1 2 . 1 2 2 3 n
2.
3. a n = n ( 3n 1) 4. If the n-th partial sum of a series
( n!) an = ( 2n )!
2
a
n =1
Homework 6
due on Monday, 30 March 2009 by 12:50 PM 1. Determine whether the integral x 3 /2 dx converges. x sin x 0 Since sin x x we see that the integrand is positive. Using limit comparison with the function x3/2 we see that LR LR x3/2 x sin x 1 cos x
Homework 6
due on Monday, 30 March 2009 by 12:50 PM
/2
1. Determine whether the integral
0 1
x 3 /2 dx converges. x sin x
2. Find lim+ x
x0
x
cos t dt. t2
3. Find all a for which the integral
1
x3
ax2 2 + 5 3x
dx converges.
4. For what values of p and q
Due on March 2, 2009 up to 12:50 p.m. Homework 3 for Math 112 1. Let f be a positive increasing continuous function on the interval [a, b] where a < b are positive real numbers. Show that
b a
f (x) dx +
f (a)
2
f (b)
2xf 1 (x)dx = b f (b)
2
a f (a) ,
2