Use the Comparison Test to show that the integral, x^a/((x^b)+1) from 1 to infinity, converges if ba+1

Question

Use the Comparison Test to show that the integral, x^a/((x^b)+1) from 1 to infinity, converges if b>a+1

anonymous · Answer

Use the Comparison test to show that the integral $$ \int\limits_{1}^{\infty}x ^{a}\div x^{b}+1 dx$$ converges if $$b> a+1$$

anonymous · Answer

compare it to 1/x^p, where p>1, you know that because b > a+1, your power in the denominator is greater than your power in the numerator, so when you go to evaluate the integral, the lim as x -> infinity for 1/infinity will be 0 which allows it to converge.