Radius of convergence for a given sum

Question

What is a brief description of the radius of convergence? How do you find the radius of convergence for $$\sum_{i=1}^{\infty}2^i\cdot x^{-3(i-1)}$$

Answer 1

Root test gives $$\lim_{n\to\infty}\sqrt[n]{\left|2^n x^{-3(n-1)}\right |} < 1$$ which can be simplified into $$2 |x^{-3}| < 1$$ from where you get $$|x|>\sqrt[3]2$$

Answer 2

I would say that it is a geometric series. Quotient = $\frac{2}{x^3}$. Konvergens when $\frac{2}{|x^3|}<1$.

Answer 3

Hint

Start rewriting $$S=\sum_{i=1}^{\infty}2^i\cdot x^{-3(i-1)}=x^3~\sum_{i=1}^{\infty} \Big(\frac{2}{x^3}\Big)^i$$ in which you recognize the sum of a geometric progression. Using the standard formula, you then have $$S=\frac{2 x^3}{x^3-2}$$ I am sure that you can take from here.