Infinite series, Geometric series

Question

If $S_n$ and $S_\infty$ denote respectively the sum of the first $n$ terms and the sum to infinity of the series $1 + 1/2 + 1/4 + 1/8 + \ldots$, find the least value of $n$ such that $|S_n − S_\infty| < 0.001$.

Answer 1

There are two well known formulas for starters:

$$S_\infty=1+r+r^2+r^3+\dots=\frac1{1-r}$$

$$S_n=1+r+r^2+r^3+\dots+r^n=\frac{1-r^{n+1}}{1-r}$$

Here we have $r=\frac12$.

$$S_\infty=1+r+r^2+r^3+\dots=\frac1{1-\frac12}=2$$

$$S_n=1+r+r^2+r^3+\dots+r^n=2-\left(\frac12\right)^n$$

Subtracting the two, we have

$$|S_\infty-S_n|=\left(\frac12\right)^n<0.001$$

$$n\ge\lceil\log_{1/2}(0.001)\rceil$$

$$n\ge10$$