The following is an alternative series representation of the Riemann zeta function.
$$\zeta(s)=\frac{1}{1-2^{1-s}}\sum\frac{(-1)^{n+1}}{n^{s}}$$
Unlike the representation $\zeta(s)=\sum 1/n^s$, the above representation is valid for $\sigma>0$ where $s=\sigma+it$.
Question: Is there a simple proof that this representation can have zeros?
To be clear, I don't need to show where the zeros are, just that it is possible for this representation to converge to zero for some $s$.
By simple, I mean suitable for an pre-university or early undergraduate audience.