How to prove : $\int\limits_{1}^{+\infty} x.f(x)dx$ is convergent

Question

How to prove : If $\int\limits_{1}^{+\infty} x.f(x)dx$ is convergent then $\int\limits_{1}^{+\infty} .f(x)dx$ is too convergent.

Answer 1

Landscape made a great suggestion in the comments to use Dirichlet's test, so I thought I would write up the answer in this community wiki.

Dirichlet's test says that $\int_c^\infty \phi(x)\psi(x)$ converges if

1. $\phi(x) \to 0$
2. $\int_{c}^\infty |\phi'(x)|\,dx < \infty$, and
3. $\int_{c}^x \psi(t)\, dt$ is bounded as $x \to \infty$.

Here we can let $\psi(x) = xf(x)$ and $\phi(x) = 1/x$. Note that $$\int_{1}^\infty |\phi'(x)|\,dx =\int_{1}^\infty \frac1{x^2} \,dx < \infty.$$

Note: to prove Dirichlet's test, just integrate by parts.