hello. i have to test for convergence and discuss if its absolte convergent or not:
$\displaystyle \sum \frac{(-1)^n} {\sqrt {n}}$
I think i should use the Leibniz Rule, but not sure how to apply it here...
Hi
$\displaystyle \displaystyle\sum_{k=1}^{\infty} |\frac{(-1)^{k}}{\sqrt{k}}| =$ $\displaystyle \displaystyle\sum_{k=1}^{\infty} \frac{1}{\sqrt{k}} $
And on this series you could use the integral test on $\displaystyle h(x) = \frac{1}{\sqrt{x}} $
And this series is divergent because the integral is divergent.
But $\displaystyle \displaystyle\sum_{k=1}^{\infty} \frac{(-1)^{k}}{\sqrt{k}}$ is convergent by Leibniz convergene theorem.
Because the series is alternating, decreasing and $\displaystyle \lim_{n\to \infty} a_{n} = 0 $