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Integral and series questions 
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#1
May107, 06:10 PM

P: 3

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/(n1)! 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 straigtforward 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? 


#2
May107, 06:18 PM

Emeritus
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PF Gold
P: 16,091

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. 


#3
May207, 09:48 AM

P: 3

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 teh 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. 


#4
May207, 10:08 AM

Emeritus
Sci Advisor
PF Gold
P: 4,500

Integral and series questions
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 


#5
May207, 03:39 PM

P: 3

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.) 


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