1. ## Comparison Test Proof

This is my assignment for analysis. When I tried looking more into this theorem, nothing online seems to state this test in a similar way. I don't really understand what the proof is telling me, so I am having a hard time proving it to be true.

Let g:[a, ∞ ) ->R be a nonnegative function satisfying g is Riemann integrable on [a,c] for every c>a and the integral from a to ∞ g(x) dx <∞ . If f: [a, ∞ ) statisfies
a) f is Riemann integrable for every c>a and
b) |f(x)|<= g(x) for all x in [a, ∞ )
then the improper integral of f on [a, ∞ ) converges and |integral from a to ∞ f(x)|<= the integral from a to ∞ g(x)

Any help explaining this theorem or starting this proof would be much appreciated.

2. ## Re: Comparison Test Proof

Originally Posted by renolovexoxo
Let g:[a, ∞ ) ->R be a nonnegative function satisfying g is Riemann integrable on [a,c] for every c>a and the integral from a to ∞ g(x) dx <∞ . If f: [a, ∞ ) statisfies
a) f is Riemann integrable for every c>a and
b) |f(x)|<= g(x) for all x in [a, ∞ )
then the improper integral of f on [a, ∞ ) converges and |integral from a to ∞ f(x)|<= the integral from a to ∞ g(x)

If $\displaystyle f$ is integrable on $\displaystyle [a,c]$ then $\displaystyle |f|$ is integrable on $\displaystyle [a,c]$ and $\displaystyle \left| {\int_a^c {f(x)dx} } \right| \leqslant \int_a^c {\left| {f(x)} \right|dx} \leqslant \int_a^c {g(x)dx}$.

3. ## Re: Comparison Test Proof

How could I start proving that though?

4. ## Re: Comparison Test Proof

Originally Posted by renolovexoxo
How could I start proving that though?

Are you saying that your class has not proved the following?

If $\displaystyle f$ is integrable on $\displaystyle [a,c]$ then $\displaystyle |f|$ is integrable on $\displaystyle [a,c]$ and $\displaystyle \left| {\int_a^c {f(x)dx} } \right| \leqslant \int_a^c {\left| {f(x)} \right|dx}.$

If not, then I don't know how to do this problem.
You must prove that first.

5. ## Re: Comparison Test Proof

We haven't yet, that might why explain why I have no idea how to start. It's due soon though, so any help would be appreciated. I understand how that follows from the triangle inequality.