Let $r>0.$ In each problem below, answer the question "Is there a holomorphic $f$ in $D(0,r)$ that satisfies the given condition for infinitely many $n \in \mathbb{N}?$"
$f(\frac{1}{n}) = \frac{1}{n^2}$
$f(\frac{1}{n}) = \frac{1}{n^2-1}$
$|f(\frac{1}{n})| \le \frac{1}{2^n}$
$f(\frac{1}{n}) = \frac{(-1)^{n+1}}{n}$
I have a hunch that the following theorem should be useful here, but I'm not sure how it should be used: if two holomorphic functions $f$ and $g$ on a domain $D$ agree on a set $S$ which has an accumulation point $c$ in $D$ then $f = g$ on $D$.
Hints: 1. @juan arroyo has given you the heads-up.
Think of it as $f(z) = \frac {1}{(1/z)^2-1}$ for certain values of $z.$
$f\equiv 0$ comes to mind. You'll have a more interesting problem if "nonconstant" is part of the requirement.
Think about $n$ odd.