# How do you determine if the series the converges conditionally, absolutely or diverges given Sigma ((-1)^(n+1))/(n^1.5) from [1,oo)?

Jan 12, 2017

The series:

${\sum}_{n = 1}^{\infty} {\left(- 1\right)}^{n + 1} / {n}^{\frac{3}{2}}$

is absolutely convergent

#### Explanation:

Given the series:

(1) ${\sum}_{n = 1}^{\infty} {\left(- 1\right)}^{n + 1} / {n}^{\frac{3}{2}}$

we can test for convergence the series:

(2) ${\sum}_{n = 1}^{\infty} \left\mid {\left(- 1\right)}^{n + 1} / {n}^{\frac{3}{2}} \right\mid = {\sum}_{n = 1}^{\infty} \frac{1}{n} ^ \left(\frac{3}{2}\right)$

If this series converges, then the series (1) converges absolutely ( and also conditionally ).

We can apply the integral test to the series (2) using the function:

$f \left(x\right) = \frac{1}{x} ^ \left(\frac{3}{2}\right)$

As, for $x \in \left[1 , + \infty\right)$, $f \left(x\right)$ is positive and decreasing, it is infinitesimal for $x \to + \infty$ and $f \left(n\right) = \frac{1}{n} ^ \left(\frac{3}{2}\right)$.

Se we calculate:

${\int}_{1}^{\infty} \frac{\mathrm{dx}}{x} ^ \left(\frac{3}{2}\right) = {\left[- \frac{2}{x} ^ \left(\frac{1}{2}\right)\right]}_{1}^{\infty} = 2$

which means that the series (1) is absolutely convergent.