I have been trying to prove or disprove for a few weeks (to avail) whether or not multiplying by a sufficiently smooth cutoff will analytically continue a function up until it hits a singularity on the real line (instead of only converging in a radius equal to the distance to the nearest singularity in the complex plane). For instance, one can analytically continue $\sum_{n=0}^\infty x^n$ by using a cutoff function and allowing $d_t$ to go to zero as the number of terms increases. 
The closest thing I have seen so far is from Tao's blog (https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/), talking about how sufficiently smooth cutoff functions can assign values to divergent series, but I would like to know whether that this value it assigns agrees with the analytic continuation of functions with a non-zero radius of convergence.
So far, I have been able to prove that this polynomial will converge to the analytic continuation of a function: $$\lim_{\varepsilon \to 0^{-+}}\sum_{a=0}^{\frac{R}{\varepsilon}}\frac{f^{(n)}(0)}{n!}\prod_{k=0}^{a-1}\left(x+\varepsilon k\right)$$ (whether $\varepsilon$ approaches 0 from the positive or negative direction depends on which direction the singularity is on the real line). From testing, it appears that this polynomial does apply a transformation that is a smooth cutoff to the terms, but I haven't been able to prove that any smooth cutoff will work, only this particular cutoff.
Are there proofs out there that show whether or not any smooth cutoff function will work?
Edit: I'd like to add that the behavior in the complex plane of these type of functions is very interesting. I'd love any insight on why the convergence looks the way it does. This is what the converge looks like when applying a similar smooth cutoff function to the one applied in the Desmos graph. Anywhere that is blue means the function diverges from the analytic continuation. Singularities are the black circles
As for the anlytic continuation function I have, it also seems to behave interestingly based on the direction it is pointed towards (one can generalize the function by viewing epsilon an imaginary number whose length goes to 0). Here is an example of what that looks like (https://i.stack.imgur.com/N9iCa.jpg).
For the function $\frac{1}{1-x}$, the convergence looks like this 
