How do I use the comparison test?

Question

Hi, I get the idea that if something is larger than a diverging function, it also diverges and vice versa but my textbook gives me this problem

∫ 0 to infinity of dx/(1+x²) and the solution key says to use the comparison test? What function am I supposed to compare it to. For me, I compared it to dx/x² which diverges and the function is less, so doesn’t the test tell you nothing?

Answer 1

your using the right parent function to compare it to 1/(x^2). but as x goes to infinity then the entire function will tend to 0. The comparison test tells u to look at ur original function 1/(1+x^2) and determine if it is larger than the parent function or less than the parent function. in our case

1/(1+x^2) < 1/(x^2) so the if 1/(x^2).converges then so must the baby function 1/(1+x^2) converge.

if the baby function were larger than the parent function say

1/x > 1/(x^2) then the comparison test would be invalid, which is not the case here