Question

Are these integrals divergent of convergent?

Answer 1

\[\int\limits_{2}^{\infty} \frac{ 1 }{ 1+x^2 }\]
and
\[\int\limits_{1}^{\infty}\frac{ 1 }{ x }\]

Answer 2

Are they both convergent?

Answer 3

\[\frac{1}{1+x^2}<\frac{1}{x^2}\implies\int_2^\infty\frac{\mathrm{d}x}{1+x^2}<\int_2^\infty\frac{\mathrm{d}x}{x^2}\]You should be able to compute the latter integral and make a conclusion from there.

For the second integral: are you allowed to use the antiderivative for \(\dfrac{1}{x}\) and the fundamental theorem of calculus? Or are you supposed to use a purely analytical approach?

Answer 4

1/x is always divergent.

First should converge.

You can tell by the p-test... integral 1/x^p converges as long as p > 1

Answer 5

Analytical Approach. But i get it now. Thanks!