Question

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

Answer 1

Let

`a_n={n(x+2)^n}/{3^{n+1}}`. `Rightarrow a_{n+1}={(n+1)(x+2)^{n+1}}/{3^{n+2}}`

By Ratio Test,

`lim_{n to infty}|{a_{n+1}}/{a_n}|=lim_{n to infty}|{(n+1)(x+2)^{n+1}}/{3^{n+2}}cdot{3^{n+1}}/{n(x+2)^n}|`

by cancelling out common factors,

`=lim_{n to infty}|{(n+1)(x+2)}/{3n}|`

by pulling `{|x+2|}/3` out of the limit and simplifying a bit,

`={|x+2|}/3lim_{n to infty}|1+1/n|={|x+2|}/3<1`

by multiplying by 3,

`Rightarrow |x+2|<3=R`

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

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I hope that this was helpful.