Does $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}-\frac{2}3}$ converge or diverge

Question

Does this series converge or diverge?

$$\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}-\frac{2}3}$$

I tried using the limit comparison test with $\frac{1}{\sqrt{n}}$, which diverges.

$$\lim_{n\to\infty}{\frac{{\sqrt{n}}}{\sqrt{n}-\frac{2}3}}=1$$

Then the series diverges, is this right or I'm wrong?

Answer 1

Yes it is absolutely right, indeed note that

$$\dfrac{1}{\sqrt{n}-\frac{2}3}\sim \dfrac{1}{\sqrt{n}}$$

and the latter diverges for p test.

As an alternative by direct comparison test

$$\dfrac{1}{\sqrt{n}-\frac{2}3}\ge \dfrac{1}{\sqrt{n}}$$