Filter to speed up convergence of geometric series
The problem is that in the case with positive $c$ you have a DC desired signal ($lim$) and a low frequency noise ($c^n$). In the case when $c$ is negative you have a high frequency noise signal, which is of course much easier to separate from the desired DC value.

If you figure out some way to estimate $c$ (sign and magnitude) you can use a simple FIR filter


b=[1,-c]/(1-c); a=1;


which will exactly eliminate the exponentially decaying noise:

$$y[n]=\frac{1}{1-c}(x[n]-cx[n-1])=\frac{1}{1-c}(lim+c^n-c\cdot lim - c\cdot c^{n-1})=lim$$