Q:

Limit of Geometric Sequence when $r\approx 1$

We know that the Geometric sequence $r^n$ converges for $|r|<1$ and converges to 1 for $r=1$.
Suppose that $r\approx 1$ say $r=0.99$. We know that the sequence will converge to $0$ but my question is can we increase the speed of convergence? I am facing such issues while coding and I want to increase the speed of convergence. As in coding, I can't take $n\rightarrow \infty$ I want some value which works for me. For, $r\in [0,0.8]$ or any such interval I am able to work with $n=100$ but not when $r\approx 0.99$

A:

If you have a set precision $0<\epsilon<1$, for given $0< r <1$ you can find $n$ such as $r^n<\epsilon$. Here is how: $\large{\ln r^n< \ln \epsilon}$ means than $\large{n >\frac{\ln \epsilon}{\ln r}}$ (sign changes because we divide by a negative number).

