Numero di Eulero – Serie

Calcolo approssimato di e con la serie

$\displaystyle e = \sum_{n=0}^{+\infty} \frac{1}{n!}$

= $\displaystyle \frac{1}{0!} + \displaystyle \frac{1}{1!} + \displaystyle \frac{1}{2!} + \displaystyle \frac{1}{3!} + \displaystyle \frac{1}{4!} + \displaystyle \frac{1}{5!} + ...$

= $1 + 1 + \displaystyle \frac{1}{2} + \displaystyle \frac{1}{6} + \displaystyle \frac{1}{24} + \displaystyle \frac{1}{120} + ...$

La serie converge al valore di e molto… velocemente: si ottengono 8 cifre esatte dopo pochi passi!

Osserva i valori esatti in 4° colonna

n	n!	$\displaystyle a_n = \frac{1}{n!}$	$\displaystyle S_n = \sum_{i=0}^n a_i$	…
0	1	1	1	= 1,0
1	1	1	2	= 2,0
2	2	$\displaystyle \frac{1}{2}$	$\displaystyle \frac{5}{2}$	= 2,5
3	6	$\displaystyle \frac{1}{6}$	$\displaystyle \frac{8}{3}$	= 2,666…
4	24	$\displaystyle \frac{1}{24}$	$\displaystyle \frac{65}{24}$	= 2,0708333…
5	120	$\displaystyle \frac{1}{120}$	$\displaystyle \frac{163}{60}$	= 2,71666…
6	720	$\displaystyle \frac{1}{720}$	$\displaystyle \frac{1957}{720}$	= 2,7180555…
7	5040	$\displaystyle \frac{1}{5040}$	$\displaystyle \frac{685}{252}$	= 2.718253968…
8	40320	$\displaystyle \frac{1}{40320}$	$\displaystyle \frac{109601}{40320}$	= 2.71827876984…
9	362880	$\displaystyle \frac{1}{362880}$	$\displaystyle \frac{98641}{36288}$	= 2.718281525573…
10	3628800	$\displaystyle \frac{1}{3628800}$	$\displaystyle \frac{9864101}{3628800}$	= 2.718281801146…