Solving Integral Tests and Constructing Continuous Monotone Functions

[kathrynag]

Integration
1.Let f be integrable on [a; b] for every b > a; where a is fixed. Define

integralf(x)dx(bonds on integral(a,infinity) = limb-->infinity(integralf(x)dx)(bounds a,b);

provided the limit exists.

Prove the so called integral test: if f(x) >= 0 and if f decreases monotonically for x >=1, then integral f(x)dx(bounds 1,infinity) converges (to a finite number) if and only if sum(f(n)converges.

2. Let f and g be integrable functions on [a; b]: Prove

|integral(f*gdx)|^2<=integralf^2dx*integra1g^2dx

3.Assume f is integrable on [a,b] and has a jump discontinuity at c in (a,b). This means that both one sided limits exists as x approaches c from the left and right, but that they are not equal.Show that F(x)=integral(f)(bounds a,x) is not differentiable at x=c.

4. The existence of a continuous monotone function that fails to be differentiable on a dense subset of R is what this problem concerns. Show how to construct such a function using the result of the previous problem and the fact that h(x) is a monotone function defined on all of R that is continuous at every irrational point where:

h(x)=sum(u_n(x)) u_n(x)=1/2^n for x>r_n and 0 for x<=r_n

where r_n are the rational numbers



Here are my ideas so far:
1. I feel like the monotonically decreasing part will help.

2. For 2, I considered the quadratic polynomial in the variable t defined by P(t) =integral(f + tg)^2. I squared this all out which worked out fine and I feel like this will help me, but I don't know what to with the t's.

3. I feel like the answer comes from the fact that the limits are not equal and I use that somehow.

4. Since I haven't figured out 3, have no clue.

 

[Char. Limit]

Integration
1.Let f be integrable on [a; b] for every b > a; where a is fixed. Define

$$\int_a^{\infty} f(x) dx = \lim_{b\to\infty} \int_a^b f(x) dx$$

provided the limit exists.

Prove the so called integral test: if f(x) >= 0 and if f decreases monotonically for x >=1, then $$\int_1^{\infty} f(x) dx$$ converges (to a finite number) if and only if $$\sum_{n=1}^\infty f(n)$$ converges.

2. Let f and g be integrable functions on [a; b]: Prove

$$\left|\int f*g dx\right|^2 \leq \left(\int f^2 dx\right) \left(\int g^2 dx\right)$$

3.Assume f is integrable on [a,b] and has a jump discontinuity at c in (a,b). This means that both one sided limits exists as x approaches c from the left and right, but that they are not equal.Show that $$F(x)=\int_a^x f(\xi) d\xi$$ is not differentiable at x=c.

4. The existence of a continuous monotone function that fails to be differentiable on a dense subset of R is what this problem concerns. Show how to construct such a function using the result of the previous problem and the fact that h(x) is a monotone function defined on all of R that is continuous at every irrational point where:

$$h(x) = \sum u_n(x), \\ u_n(x) = \frac{1}{2^n} \\ for \\ x>r_n \\ and \\ 0 \\ for \\ x\leq r_n$$

where r_n are the rational numbers
Here are my ideas so far:
1. I feel like the monotonically decreasing part will help.

2. For 2, I considered the quadratic polynomial in the variable t defined by P(t) =integral(f + tg)^2. I squared this all out which worked out fine and I feel like this will help me, but I don't know what to with the t's.

3. I feel like the answer comes from the fact that the limits are not equal and I use that somehow.

4. Since I haven't figured out 3, have no clue.

I added in some latex. Feel free to tell me if I interpreted it correct.

 

[kathrynag]

Thanks a lot! I was posting from my Ipod so I couldn't get the Tex.

 

[kathrynag]

For 2. I looked at integral(f+tg)^2
integral(f^2+2tgf+t^2g^2).

 

[kathrynag]

Ok, think I have a good start on 1 and 2 now. Are there any hints on starting 3 and 4?

 

[kathrynag]

3. We have $$lim_{c^{-}}$$$$\neq$$$$lim_{c^{+}}$$.
Would I maybe use upper and lower sums to show not integrable?