I am trying to solve a problem where I have to find the radius of finite convergence problem. I believe that I solved the problem correctly, receiving an answer of 1. However, I was informed that this was incorrect but I am not sure why.
My process involved using the ratio test and ultimately received:
(x+2) lim n/(n+1) which goes to 1
Any help would be greatly appreciated
If you use the Ratio Test, you have $$ \begin{split} \lim_{n \to \infty} \left|\dfrac{\dfrac{(n+1)!(x+2)^{n+1}}{(n+1)^{n+1}}}{\dfrac{n!(x+2)^n}{n^n}} \right|&= \lim_{n \to \infty} \left|\dfrac{(n+1)!(x+2)^{n+1}n^n}{n!(x+2)^n(n+1)^{n+1}} \right| \\ &= \lim_{n \to \infty} \left| \dfrac{(n+1)(x+2)n^n}{(n+1)(n+1)^n} \right| \\ &= \lim_{n \to \infty} \left| \dfrac{(x+2)n^n}{(n+1)^n} \right| \\ &= \lim_{n \to \infty} \left|(x+2) \; \left( \dfrac{n}{n+1}\right)^n \right| \\ &= \lim_{n \to \infty} \left|(x+2) \; \left( \dfrac{n+1}{n}\right)^{-n} \right| \\ &= \lim_{n \to \infty} \left|(x+2) \; \left( 1+ \dfrac{1}{n}\right)^{-n} \right| \\ &= \lim_{n \to \infty} \left|(x+2) \; \left(\left( 1+ \dfrac{1}{n}\right)^{n}\right)^{-1} \right| \\ &= |x+2| \cdot e^{-1}, \end{split} $$ which is what you found. The trick with these harder Ratio Test problems is working slowly and being careful with the Algebra.