Find all 
such that the series
![\[ \sum_{n=0}^{\infty} \frac{(-1)^n x^{3n}}{n!} \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-a396cce4eed1c46902f65b0a4714a273_l3.png "Rendered by QuickLaTeX.com")
converges and compute the sum.
By the ratio test we have
![\begin{align*} \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| &= \lim_{n \to \infty} \left( \frac{x^{3n+3}}{(n+1)!} \right) \left( \frac{n!}{x^{3n}} \right) \\[9pt] &= \lim_{n \to \infty} \frac{x^3}{n+1} \\[9pt] &= 0 \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-8b9dc23ae4d4e9ff8e8f0946792062f2_l3.png "Rendered by QuickLaTeX.com")
for all 
. Thus, the series converges for all . Furthermore, we compute the sum from the series expansion of 
,
![\[ \sum_{n=0}^{\infty} \frac{(-1)(x^3)^n}{n!} = e^{-x^3}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-ef029a46af395a295fc6905227d8487f_l3.png "Rendered by QuickLaTeX.com")
Find all such that the series
converges and compute the sum.
By the ratio test we have
for all . Thus, the series converges for all . Furthermore, we compute the sum from the series expansion of ,