Proving that $\sum_ {n=1}^{\infty}{\frac{(-1)^n}{\sqrt{n}}} $ converges

Question

Prove that the following series converges:

$$\sum_ {n=1}^{\infty}{\frac{(-1)^n}{\sqrt{n}}} $$

$$\frac{1}{\sqrt{n+1}} > \frac{1}{\sqrt{n}}$$ $$\lim_{n \to \infty} \frac{1}{\sqrt{n}} = 0$$ So, the alternating series converges.

Is it right to my procedure?

Answer 1

To use Alternating series test for series $\sum(-1)^n a_n$ you should have

1. $a_n\geq 0$

2. $a_n\geq a_{n+1}$

3. $\lim a_n = 0$

In your approach you wrote that $a_{n+1} = \frac{1}{\sqrt{n+1}}>\frac{1}{\sqrt n} = a_{n}$ which is wrong and not what you need. Fix it and your solution will be correct.