I have a power series $\sum a_n z^n=\frac{2-z}{1-z}$ and I want to determine the radius of convergence of this power series.
Rewriting the expression on the RHS we see that it is equivalent to $1+\frac{1}{1-z}$ and so the power series for this expansion is $2+z+z^2+z^3....$
I now want to determine the radius of convergence of this power series using methods from real analysis such as the ratio test etc... but I cannot represent the $n^{th}$ term of this power series as a single formula because of the $2$ at the start of this power series. If I wanted to use something like the ratio test to determine if the series converges absolutely or not, would it be fine to ignore the first term, say, (or some number of finite terms) and then consider the ratio $|\frac{a_{n+1}}{a_n}|$ for $n$ beyond these finite number of terms? This would be much easier since for $n>1$ the power series is simply $a_n=z^n$ and so I could apply the ratio test and deduce that the radius of convergence $R_c=1$.
We have $a_0=2$ and $a_n=1$ for $n \ge 1$. Hence $|\frac{a_{n+1}}{a_n}|=1$ for $n \ge 1$.
Your turn !