Find a sequence of functions holomorphic on the punctured unit disc that satisfy certain properties about singularities. Let $(f_n)$ be a sequence of functions that are holomorphic on the punctured unit disc $D'=\{z\in \mathbb{C}: 0<|z|<1\}$, satisfying:
(i) For each $n\in\mathbb{N}$, the function $f_n$ has a pole of order $n$.
(ii) $\sum_{n=1}^\infty f_n$ converges to a function $f$ uniformly on each compact subset of $D'$.
Show that
a) for some choice of the sequence $(f_n)$, the function $f$ has a removable singularity at $0$. 
b) for some choice of the sequence $(f_n)$, the function $f$ has a simple pole at $0$. 
c) for some choice of the sequence $(f_n)$, the function $f$ has an essential singularity at $0$. 
I was able to do part (c) by considering the function $e^{1/z}$. For parts (a) and (b), I have been thinking of the functions $\frac{\sin z}{z}$ and $1/z$ respectively, but I cannot see how to write them as a sum of such a sequence of $(f_n)$. Any hint or advice is most welcome. Thank you.
 A: For a), think of the principal part of the partial sums. If
$$s_N(z) = \sum_{n=1}^N f_n(z) = h_N(z) + g_N(z),$$
where $g_N$ is holomorphic on the whole disk and
$$h_N(z) = \sum_{k=1}^N \frac{a_{N,k}}{z^k}$$
is the principal part, how could you choose the $f_n$ so that


*

*the principal part $h_N(z)$ is very simple for all $N$, and

*$h_N(z) \to 0$ uniformly on every compact subset $K\subset D'$?



The simplest principal part of a pole of order $N$ at $0$ has the form $\frac{c_N}{z^N}$. To achieve that simple form, the principal part of $s_N$ must be immediately cancelled by the principal part of $s_{N+1}$. So we can choose
$$f_n(z) = \frac{c_n}{z^n} - \frac{c_{n-1}}{z^{n-1}},$$
and need only ensure that $\frac{c_n}{z^n}$ converges to $0$ locally uniformly on $D'$. That is achieved whenever the sequence $(c_n)$ converges to $0$ fast enough, a famous such sequence is $c_n = \frac{1}{n!}$.
For part b), modify the function $f_1$ from part a), say add $\frac{1}{z}$ to it, and leave the others unchanged.
