What is the Limit of a Convergent Series with Increasing Upper Bound?

[pakkman]

1. If f is Riemann integrable from a to b, and for every rational number r, f(r)=0, then show that the integral from a to b of f(x) is 0.

The problem with this question is that you don't know what f is at an irrational. I know that I'm probably supposed to use that rationals are dense in R, but other than that, I'm not sure.

2. Let f(x)= sigma sin nx/(n-1)! where sigma is the sum from n=1 to infinity. Show that the int f(x)dx exists (Riemann integral is from 0 to pi), and evaluate.

So, I guess I show that the integral exists because f(x) is pointwise continuous? I'm really confused on this question, and how I can evaluate it.

3. What's a relatively straigt-forward way of proving that if f is riemann integrable, then lim n-> infinity of int f(x) cosnx dx =0, where the integral is evaluated from a to b? Any hints?

 

[Hurkyl]

1. If f is Riemann integrable from a to b, and for every rational number r, f(r)=0, then show that the integral from a to b of f(x) is 0.

The problem with this question is that you don't know what f is at an irrational. I know that I'm probably supposed to use that rationals are dense in R, but other than that, I'm not sure.

You probably also need to use the fact that f is Riemann integrable on [a, b].

2. Let f(x)= sigma sin nx/(n-1)! where sigma is the sum from n=1 to infinity. Show that the int f(x)dx exists (Riemann integral is from 0 to pi), and evaluate.

So, I guess I show that the integral exists because f(x) is pointwise continuous? I'm really confused on this question, and how I can evaluate it.

Cast rigor aside for a moment; try to evaluate this integral naively.

3. What's a relatively straigt-forward way of proving that if f is riemann integrable, then lim n-> infinity of int f(x) cosnx dx =0, where the integral is evaluated from a to b? Any hints?

The first thing I notice is that cos nx oscillates very quickly when n is big. Also, I strongly suspect you are in the "interchanging limits" section of your course...

Actually, before I did any serious work on this problem, I would do a quick search for a theorem that would prove this statement. There are a lot of useful integral theorems I cannot remember, and this smells like the kind of thing that might be proven by one of them.

 

[pakkman]

I got 3 and I got part of 2. Is sigma (-1)^n/n! (sum from 0 to infinity) -e^-1? I vaguely remember this, but not sure about the proof...

I'm still really stumped on question 1. Can someone clarify?

I'm also trying to figure out the value of lim n->infinity of sigma k/(n^2+k^2) where the sum is from k=0 to k=2n. I guess I have to do some manipulation of the summand quantity, but I'd really appreciate a hint.

 

[Office_Shredder]

For f to be Riemann integrable, the least upper bound of step functions less than f must have area equal to the greatest lower bound of step functions greater than f.

I would suggest you start by looking at functions like f(r)=0 if r is rational, and 1 if r is irrational. Try finding what the upper and lower integrals of f are there, and you'll start to get an intuitive feel for why the integral must be zero for it to exist

 

[pakkman]

For f to be Riemann integrable, the least upper bound of step functions less than f must have area equal to the greatest lower bound of step functions greater than f.

I would suggest you start by looking at functions like f(r)=0 if r is rational, and 1 if r is irrational. Try finding what the upper and lower integrals of f are there, and you'll start to get an intuitive feel for why the integral must be zero for it to exist

Thanks... it helps intuitively, but I'm having a little trouble with a formal proof. I'll keep working on it.

Can someone help me on lim n->infinity of sigma k/(n^2+k^2) where the sum is from k=0 to k=2n.? I can't seem to simplify it. I know it converges (from Matlab), and it really depends on the upper value of k (whether it's 2n, or 3n, or 4n, etc.)