Question

Use the integral Test to determine whether the series converges. I used the p-series test, so I know the series diverges;however, it asks me to prove it using the integral test. How do you that again?

Answer 1

\[\sum_{n=1}^{\infty} 4/\sqrt{n}\]

Answer 2

To apply the integral test, the sequence has to be positive and decreasing, which is true for \(\dfrac{1}{\sqrt n}\).

The test then tells you that
\[\int_1^\infty \frac{1}{\sqrt x}~dx~\text{converges/diverges}~~\Rightarrow~~\sum_{n=1}^\infty \frac{1}{\sqrt{n}}~\text{converges/diverges}\]
Convergence/divergence occurs respectively.