Your answer looks right.
It follows from the following theorem (Functions of One Complex Variable, John B Conway, Indian Edition, page number 169).
Theorem 5.12: Let $\{a_n\}$ be a sequence in $\mathbb{C}$ such that $\lim |a_n| = \infty$ and $a_n \neq 0$ for all $n \ge 1$. If $\{p_n\}$ is any sequence of integers such that
$ \sum_{n=1}^{\infty} \left(\frac{r}{|a_n|}\right)^{p_n+1} \lt \infty$
for all $r \gt 0$, then
$ f(z) = \prod_{n=1}^{\infty} E_{p_n}\left(\frac{z}{a_n}\right)$ converges and $f$ is an entire function with zeroes only at the points $a_n$. If $z_0$ occurs in $\{a_n\}$ exactly $m$ times, then $f$ has a zero $z_0$ of multiplicity $m$.
You have chosen $p_n = n$ which works.
This theorem is used to prove the Weirstrass Factorization theorem.