Radius of convergence is for power series but how does one go about computing the radius of convergence of the infinite sum $\sum_{k=1}^{\infty} \frac{k}{k+1}\left(\frac{2x+1}{x}\right)^k\ ?$
Can you find the radius $R$ directly by the standard $1/\limsup$ formula (or the Ratio Test) or do you have to make some kind of substitution to get it into the right form? I got the domain of convergence $(-1, -1/3)$ when I used the Ratio Test.
If you do need a substitution, give me a hint of how to go about that. If I am okay computing $R$ correctly with Ratio or Root Test, give me a hint why that's acceptable (even though $(2x+1)/x$ isn't a term of a polynomial).