Comparison Test for Improper Integrals — Question 3

Question 3

Use a lower comparison to determine convergence: $$\int_1^\infty\frac{2+\sin x}{x}\,dx.$$

Question 3 – Solution

Step 1: Bound the numerator. Since $\sin x\ge-1$, $$2+\sin x\ge1.$$ For $x\ge1$, divide by the positive number $x$: $$\frac{2+\sin x}{x}\ge\frac1x\ge0.$$ Step 2: Test the lower comparison. $$\int_1^\infty\frac{dx}{x} =\lim_{b\to\infty}\ln b=\infty.$$ Step 3: Apply direct comparison. The smaller integral diverges, so the original integral diverges. $$\boxed{\text{diverges}}$$