The Taylor expansion of a constant function is simply the constant followed by $0\cdot x^n$ for all subsequent $n$. But is there a power series that converges to a certain constant for all $x\in\mathbb{R}$ yet has a constant term of $0$ and does not have $0$ as a coefficient for all other powers?
Or is it possible to prove that such a series cannot exist? Otherwise, can we get an alternative result, such as a series that converges merely for almost all $x$ or just on a particular interval?
So far, I'd attempted to get a power series by simply expanding the series of $\left(\frac{\sin x}{2}\right)^p$ *. For example, $\frac{x^{10}}{1024}+\mathcal{O}(x^{12})$ and $\frac{x^{100}}{10^{30}}+\mathcal{O}(x^{102})$ for $p=10$ and $p=100$ respectively. While this is successfully approximately constant (i.e. $0$) for all real $x$, it just wipes out the smaller powers of $x$, which isn't very useful. For example, the figure below shows the partial sum to $n=10$ of the series for $\frac{\sin x}{2}$, then taken to the power of $10$. (Interactive Desmos link).
* The denominator of $2$ was to account for values of $\pm1$.