Determine whether the series is absolutely convergent, conditionally convergent, or divergent.
Determine whether the series is absolutely convergent, conditionally convergent, or divergent.
Ratio test:
$\displaystyle \lim_{k \to \infty}\left|\frac{a_{k + 1}}{a_k}\right|$
$\displaystyle = \lim_{k \to \infty}\left|\frac{\frac{(-1)^{k + 2}\cdot 1 \cdot 4 \cdot 7 \cdot \dots \cdot (3k - 2) \cdot (3k + 1)}{(k + 1)!2^{k + 1}}}{\frac{(-1)^{k + 1}\cdot 1 \cdot 4 \cdot 7 \cdot \dots \cdot (3k - 2)}{k!2^k}}\right|$
$\displaystyle = \lim_{k \to \infty}\frac{\frac{1 \cdot 4 \cdot 7 \cdot \dots \cdot (3k - 2) \cdot (3k + 1)}{(k + 1)!2^{k + 1}}}{\frac{1 \cdot 4 \cdot 7 \cdot \dots \cdot (3k - 2)}{k!2^k}}$
$\displaystyle = \lim_{k \to \infty}\frac{3k + 1}{2(k + 1)}$
$\displaystyle = \lim_{k \to \infty}\frac{3k + 1}{2k + 2}$
$\displaystyle = \lim_{k \to \infty}\frac{3}{2}$ by L'Hospital's Rule
$\displaystyle = \frac{3}{2}$
$\displaystyle > 1$.
So the series is divergent.