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.(adsbygoogle = window.adsbygoogle || []).push({});

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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Integral and series questions

**Physics Forums | Science Articles, Homework Help, Discussion**