Question

Does this converge or diverge? Which of the follow test will be used to figure out the equation?

Answer 1

From the listed choices, you can prove the series diverges using either the integral or p-series test.

• Integral test:

We have

$\displaystyle\sum_{n=1}^\infty\ge\int_1^\infty\frac{\mathrm dx}{\sqrt x}=\lim_{x\to\infty}2\sqrt x-2$

which diverges to infinity, so the series is divergent.

• p-series test:

The series

$\displaystyle\sum_{n=1}^\infty\frac1{n^p}$

converges only for $p>1$. In the given sum, we have $p=\frac12$, so the series is divergent.