Question

what is the radius of convergence for the series ((-1)^(n-1)*x^(3n)*2^n)/n

Answer 1

is it |x|<1/cubicroot(2))?

Answer 2

\[|x|<1/\sqrt[3]{2}\]

Answer 3

Let's write that one in the formal form
\[ a_n=\frac{(-1)^{n-1}x^{3n}2^n}{n} \]
Using the ratio test, we have
\[ \lim_{n \to \infty}\left| \frac{a_{n+1}}{a_n}\right| = \lim_{n \to \infty}\left| \frac{(-1)^{n}x^{3n+3}2^{n+1}}{n+1} : \frac{(-1)^{n-1}x^{3n}2^n}{n} \right|\]
Therefore,
\[ \lim_{n \to \infty}\left| \frac{a_{n+1}}{a_n}\right|=\lim_{n \to \infty}\left| \frac{2n x^3}{(n+1)}\right| =2|x|^3 \]

Now, let \[ 2|x|^3 <1\]
we have the radius of convergence is
\[ |x| < \frac{1}{\sqrt[3]{2}} \]

Answer 4

thanks just wanted to check that i did the right thing.

Answer 5

yes, you're absolutely on the right track