# Convergence of improper integrals theorems

1. Oct 23, 2013

### R_beta.v3

1. The problem statement, all variables and given/known data
I'm trying to prove these two theorems
a) if $0 \leq f(x) \leq g(x)$ for all x $\geq 0$ and $\int_0^\infty g$ converges, then $\int_0^\infty f$ converges
b) if $\int_0^\infty |f|$ converges then $\int_0^\infty f$ converges.

Obviously assuming f is integrable on every interval [0, N], N $\geq 0$.

2. Relevant equations

3. The attempt at a solution
For a).
Let F be defined by $F(x) = \int_0^x f$. $F$ is bounded above by $\int_0^\infty g$. and since, $0 \leq f(x)$ for all x, F is non-decreasing. So $\displaystyle\lim_{x\rightarrow \infty} F(x)$ exists.

For b)
$0 \leq f(x) + |f(x)| \leq 2|f(x)|$ for all x. and $\int_0^\infty 2|f|$ is convergent, so $\int_0^\infty (f + |f|)$ is convergent by (a).
The existence of $\displaystyle\lim_{x\rightarrow \infty} \int_0^x |f|$ and the existence of $\displaystyle\lim_{x\rightarrow \infty} \int_0^x (f + |f|) = \displaystyle\lim_{x\rightarrow \infty}\left( \int_0^x f + \int_0^x |f| \right)$. implies the existence of $\displaystyle\lim_{x\rightarrow \infty} \int_0^x f$

2. Oct 23, 2013