does the series $\sum | \frac{(-1)^n}{\sqrt{n}} (1+ \frac{(-1)^n}{\sqrt{n}}) |$ converge?

Question

Does the series $$S_n = \sum \bigg{|} \frac{(-1)^n}{\sqrt{n}} \left(1+ \frac{(-1)^n}{\sqrt{n}}\right) \bigg{|}$$ converge?

I could arrive at $S_n = \sum \bigg{|} \left(\frac{1}{\sqrt{n}} + \frac{(-1)^n}{n}\right) \bigg{|}$.

And I didn't know what to do from here.

Answer 1

Note that $n > \sqrt{n}$ for all $n \geq 2$. Let us rewrite the sum $S_n = \sum_{n=1}^\infty |(-1)^n||\frac{1}{\sqrt{n}}+ \frac{(-1)^n}{n}| = \sum_{n=1}^\infty |\frac{1}{\sqrt{n}}+ \frac{(-1)^{n}}{n}|$

Note that since $n > \sqrt{n}$ the expression $\frac{1}{\sqrt{n}}+ \frac{(-1)^{n}}{n}$ is positive. Thus,

$S_n= \sum_{n=2}^\infty \frac{1}{\sqrt{n}} + \frac{(-1)^n}{n} = \sum_{n=2}^\infty \frac{1}{\sqrt{n}} + L = \infty$. Note that since

Since the sum $\sum_{n=2}^\infty \frac{(-1)^n}{n} = L$ converges, and the sum $\sum_{n=2}^\infty \frac{1}{\sqrt{n}}$ does.