Calculate the radius of convergence $\sum_{n=0}^{\infty}a^{n}x^{n}$

Question

The following power series is given with $x\in\mathbb{R}$ and $a \in (0,1)$

$$\sum_{n=0}^{\infty}a^{n}x^{n}$$ Calculate its radius of convergence.

$$\lim_{n\rightarrow\infty}\frac{a^{n+1}}{a^{n}}= \lim_{n\rightarrow\infty}\frac{a^{n}\cdot a}{a^{n}}=a$$

$\Rightarrow$

$$R=\frac{1}{a}$$

So the radius of convergence of this power series is $\frac{1}{a}$.

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Did I do everything correctly?

Answer 1

Looks good!

Alternative: you may remember that $\sum x^n$ converges for $|x|<1$ so $\sum (ax)^n$ will converge for $|ax|<1$ which happens for $|x| < \tfrac{1}{|a|}$. Of course, for $0<a<1$, you have $|a|=a$.

Answer 2

$${\frac {1}{1-x}}=\sum _{n=0}^{\infty }x^{n}\quad {\text{ for }}|x|<1\!$$ let $$x\rightarrow ax$$ $${\frac {1}{1-ax}}=\sum _{n=0}^{\infty }(ax)^{n}\quad {\text{ for }}|ax|<1\!$$ so $$|x|<\frac{1}{a}$$