I am currently trying to solve #3 here (taken from Lang's Complex Analysis):
I am not seeking the answer, I just have a few questions:
- Since $\alpha_n = 1 - \frac{1}{n^2}$, $|\alpha_n|$ is redundant, and I can simplify $\frac{|\alpha_n|}{\alpha_n}$ to be 1, correct?
- What does $\overline{\alpha}_n$ mean? I thought it might be complex conjugate, but $\alpha_n$ is not complex.
- Why do we need to know that $0<x<1$?
To your first question, you're correct, since $\alpha_n$ is always real in this case, you can simplify $\frac {|\alpha_n|}{\alpha_n}$ to one
To your second question, you're also correct that $\bar {\alpha_n}$ denotes the complex conjugate, since $\alpha_n$ is real, the complex conjugate of $\alpha_n$ is just $\alpha_n$
To your third question, $0<x<1$ is important because it restricts the domain to values of $x$ where $f$ is defined