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1/factorial(n)

Sum of series 1/factorial(n)



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The solution

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  oo    
 ___    
 \  `   
  \   1 
   )  --
  /   n!
 /__,   
n = 1   
$$\sum_{n=1}^{\infty} \frac{1}{n!}$$
Sum(1/factorial(n), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{1}{n!}$$
It is a series of species
$$a_{n} \left(c x - x_{0}\right)^{d n}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}$$
In this case
$$a_{n} = \frac{1}{n!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\left(n + 1\right)!}{n!}}\right|$$
Let's take the limit
we find
False

False
The rate of convergence of the power series
The answer [src]
-1 + E
$$-1 + e$$
-1 + E
Numerical answer [src]
1.71828182845904523536028747135
1.71828182845904523536028747135
The graph
Sum of series 1/factorial(n)
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