Interval of Convergence: Find Series

[Anewk]

How would I find the interval of convergence for the following series:

i) \(\displaystyle \sum \frac{(x+2)^n}{n^2}\)

ii) \(\displaystyle \sum \frac{(-1)^kk^3}{3^k}(x-1)^{k+1}\)

iii) \(\displaystyle \sum (1+\frac{1}{n})^nx^n\)

Reason for edit: My second series was not displaying properly

 

[MarkFL]

I would ask that you edit the second expression so that it renders as you want, and then show us what you have tried so we know where you are stuck and can offer better help. :D

 

[Anewk]

I would ask that you edit the second expression so that it renders as you want, and then show us what you have tried so we know where you are stuck and can offer better help. :D

Sorry about that. Done. btw big Rush fan myself :)

 

[Prove It]

How would I find the interval of convergence for the following series:

i) \(\displaystyle \sum \frac{(x+2)^n}{n^2}\)

ii) \(\displaystyle \sum \frac{(-1)^kk^3}{3^k}(x-1)^{k+1}\)

iii) \(\displaystyle \sum (1+\frac{1}{n})^nx^n\)

Reason for edit: My second series was not displaying properly

The ratio test states that for any series $\displaystyle \begin{align*} \sum_{\textrm{all }n} a_n \end{align*}$ is convergent if $\displaystyle \begin{align*} \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1 \end{align*}$ and divergent where $\displaystyle \begin{align*} \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| > 1 \end{align*}$. The test is inconclusive if the limit is 1.

So in each of these, evaluate $\displaystyle \begin{align*} \left| \frac{a_{n+1}}{a_n} \right| \end{align*}$, evaluate its infinite limit, set it less than 1, and solve for x.