determine the radius of convergence of the following series: \n sum_{n = 0}^{infty}\frac{x^{n}}{15^{n}} \n r=

Explanation:

Step1: Apply ratio - test formula

Let $a_n=\frac{x^n}{15^n}$. Then $a_{n + 1}=\frac{x^{n+1}}{15^{n + 1}}$. Calculate $\lim\limits_{n\rightarrow\infty}\left|\frac{a_{n+1}}{a_n}\right|$. [ \begin{align*} \lim\limits_{n\rightarrow\infty}\left|\frac{a_{n + 1}}{a_n}\right|&=\lim\limits_{n\rightarrow\infty}\left|\frac{\frac{x^{n+1}}{15^{n+1}}}{\frac{x^n}{15^n}}\right|\ &=\lim\limits_{n\rightarrow\infty}\left|\frac{x^{n+1}}{15^{n+1}}\cdot\frac{15^n}{x^n}\right| \end{align*} ]

Step2: Simplify the limit expression

[ \begin{align*} \lim\limits_{n\rightarrow\infty}\left|\frac{x^{n+1}}{15^{n+1}}\cdot\frac{15^n}{x^n}\right|&=\lim\limits_{n\rightarrow\infty}\left|\frac{x^{n+1 - n}}{15^{n+1 - n}}\right|\ &=\lim\limits_{n\rightarrow\infty}\left|\frac{x}{15}\right|\ &=\left|\frac{x}{15}\right| \end{align*} ]

Step3: Find the radius of convergence

For the series to converge, $\left|\frac{x}{15}\right|<1$. Solving for $|x|$, we get $|x|<15$. The radius of convergence $R$ is given by the formula for the inequality $|x - c|<R$ (in the case of a power - series centered at $c = 0$). So $R = 15$.

Answer:

$15$