radius of convergence

for power series

$$\sum _n^{\infty }{a}_n$$

like the content in Calculus II, do the ratio test

$$\underset{n\to \infty }{\lim }|\frac{a_{n+1}}{a_n}|$$

then the radius of convergence is $\frac{1}{N}$ where $N$ is at most what $x$ can let it less than $1$ (convergence condition).

Example

$$\sum _{n=0}^{\infty }{2}^n{x}^n$$

So

$$a_n=2^n{x}^n$$

and

$$a_{n+1}=2x{2}^n{x}^n$$

then

$$\underset{n\to \infty }{\lim }|\frac{a_{n+1}}{a_n}|=|2x|$$

Then we can say the radius of convergence is $R=\frac{1}{N}=\frac{1}{2}$. $\square$