Why does the series converge?

• May 4th 2011, 12:40 PM
Pranas
Why does the series converge?
Hello.

How to prove that the series converge?

$\displaystyle $S = \sum\limits_{n = 2}^{ + \infty } {{{( - 1)}^n} \cdot {c_n}} = \sum\limits_{n = 2}^{ + \infty } {\frac{{{{( - 1)}^n}}}{{\sqrt n + {{( - 1)}^n}}}}$$

That's obviously an alternating series because $\displaystyle ${c_n} > 0$$

In addition $\displaystyle $\mathop {\lim }\limits_{n \to \infty } {c_n} = 0$$

But $\displaystyle ${c_{n + 1}}$$ is NOT always less than $\displaystyle ${c_n}$$

and apparently it prevents me from using Leibniz's test.

Without subtractions $\displaystyle $\sum\limits_{n = 2}^{ + \infty } {{c_n}}$$ diverges so I've got no way to go :(

Am I missing something obvious?
• May 4th 2011, 12:46 PM
girdav
You know that the series $\displaystyle \sum_{n=2}^{+\infty}\frac{(-1)^n}{\sqrt n}$ is convergent. Hence $\displaystyle S$ is convergent if and only if $\displaystyle \sum_{n=2}^{+\infty}(-1)^nc_n-\frac{(-1)^n}{\sqrt n}$ is convergent.