Meromorphic function analytic at $0$

Question

Let $f\colon \mathbb{C}\rightarrow\mathbb{C}$ be a meromorphic function analytic at $0$ such that $f\big(\frac{1}{n}\big)=\frac{n}{2n+1}$ for all $n\geq1$. Show:

$a)\hspace{0.25cm} f(2)=\frac{1}{4}$

$b)\hspace{0.25cm} f$ has a simple pole at $z=-2$.

I'm unsure how to proceed. I tried writing the power series expansion around $0$ but really get anywhere. I also tried writing $f$ as a rational function but that didn't help either. Any hints are appreciated.

Answer 1

Observe that $$ f\left(\frac{1}{n}\right)=\frac{1}{2+\frac{1}{n}} $$ and as $f$ is analytic at $z=0$, then the Identity Principle applies and hence $$ f(z)=\frac{1}{2+z}. $$