Estimating the Value of a Series — Question 5

*Question 5 For $\displaystyle S=\sum_{n=0}^{\infty}(1/3)^n$, define $S_N=\sum_{n=0}^{N}(1/3)^n$.

1. Compute the tail $R_N=S-S_N$ exactly.

2. Compare it with the standard geometric-tail estimate based on the first omitted term.

*Question 5 – Solution

Because $S_N$ ends at index $N$, the tail begins at $n=N+1$: $$R_N=\sum_{n=N+1}^{\infty}\left(\frac13\right)^n.$$

$$R_N=\frac{(1/3)^{N+1}}{1-1/3}=\frac{1}{2\,3^N}.$$ Thus the exact error is $\boxed{R_N=3^{-N}/2}$.

For a geometric tail with ratio magnitude $r<1$, $$|R_N|\le\frac{|a_{N+1}|}{1-r}.$$ Here $a_{N+1}=3^{-(N+1)}$ and $r=1/3$, producing $1/(2\,3^N)$. Because every term is positive and the ratio is exactly constant, the usual “bound” equals the exact tail.