# What is the radius of convergence of the series sum_(n=0)^oo(n*(x+2)^n)/3^(n+1)?

Oct 20, 2014

Let

${a}_{n} = \frac{n {\left(x + 2\right)}^{n}}{{3}^{n + 1}}$. $R i g h t a r r o w {a}_{n + 1} = \frac{\left(n + 1\right) {\left(x + 2\right)}^{n + 1}}{{3}^{n + 2}}$

By Ratio Test,

${\lim}_{n \to \infty} | \frac{{a}_{n + 1}}{{a}_{n}} | = {\lim}_{n \to \infty} | \frac{\left(n + 1\right) {\left(x + 2\right)}^{n + 1}}{{3}^{n + 2}} \cdot \frac{{3}^{n + 1}}{n {\left(x + 2\right)}^{n}} |$

by cancelling out common factors,

$= {\lim}_{n \to \infty} | \frac{\left(n + 1\right) \left(x + 2\right)}{3 n} |$

by pulling $\frac{| x + 2 |}{3}$ out of the limit and simplifying a bit,

$= \frac{| x + 2 |}{3} {\lim}_{n \to \infty} | 1 + \frac{1}{n} | = \frac{| x + 2 |}{3} < 1$

by multiplying by 3,

$R i g h t a r r o w | x + 2 | < 3 = R$

Hence, the radius of convergence $R$ is $3$.

I hope that this was helpful.