In Rudin's Principle's of Mathematical Analysis, Rudin days that we can estimate how fast the series [itex]\sum\frac{1}{n!}[/itex] converges by the following:(adsbygoogle = window.adsbygoogle || []).push({});

Put

$$

s_{n}=\sum_{k=0}^{n}\frac{1}{k!}.

$$

Then

$$

e-s_{n}

=\frac{1}{(n+1)!}+\frac{1}{(n+2)!}+\frac{1}{(n+3)!}+\cdots<\frac{1}{(n+1)!}\left(1+\frac{1}{n+1}+\frac{1}{(n+1)^{2}}+\cdots\right)=\frac{1}{n!n}

$$

so that

$$

0<e-s_{n}<\frac{1}{n!n}.

$$

The part that bothers me is

$$

\frac{1}{(n+1)!}\left(1+\frac{1}{n+1}+\frac{1}{(n+1)^{2}}+\cdots\right)=\frac{1}{n!n}.

$$

Using Maple I was able to see that

$$

\frac{1}{(n+1)!}\sum_{k=0}^{\infty}\frac{1}{(n+1)^{k}}=\frac{1}{n!n}

$$

but what if I did not have access to anything like Maple or Mathematica. How would I be able to figure out that the equality holds?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Rudin's explanation of how rapid the series 1/(n!) converges

Loading...

Similar Threads for Rudin's explanation rapid |
---|

I Proof that p is interior if p is not limit of complement |

**Physics Forums | Science Articles, Homework Help, Discussion**