Comparing remainder(residual) Rn(x)

2018-04-04 20:04:05

Can you explain to me in detail if possible why $R_n$ of $(1)$ is faster than $R_n$ of $(2)$?

$(1):$ For $$\sum_{n=0}^{\infty}\frac{1}{n!}=e$$

$$Rn=\sum_{k=n+1}^{\infty}\frac{1}{k!}=\frac{1}{(n+1)!}(1+\frac{1}{n+2}+\frac{1}{(n+2)(n+3)}+...)<\frac{1}{(n+1)}(1+\frac{1}{n+1}+\frac{1}{(n+1)^2}+...)$$

$(2):$For $$\sum_{n=0}^{\infty}x^n, |x|<1$$

$$Rn=\frac{x^{n+1}}{1-x}$$