Evaluate the improper integral if it converges, or state that it diverges.
E) diverges

Question

Evaluate the improper integral if it converges, or state that it diverges. 
E) diverges

Quizplus · Accepted Answer

The integral is of the form $\int_1^{\infty} \frac{e^x}{x^2} dx$. We can use the limit comparison test with $\frac{1}{x^2}$ as follows:
$$\lim_{x	o \infty} \frac{\frac{e^x}{x^2}}{\frac{1}{x^2}} = \lim_{x	o \infty} e^x = \infty$$
Since this limit is divergent, and $\int_1^{\infty} \frac{1}{x^2} dx$ converges (since it is a $p$-series with $p>1$), by the limit comparison test, $\int_1^{\infty} \frac{e^x}{x^2} dx$ also diverges. Therefore, the answer is (E) diverges.

Evaluate the Improper Integral If It Converges, or State That