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1/(n+1)!

Sum of series 1/(n+1)!



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The solution

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  oo          
 ___          
 \  `         
  \      1    
   )  --------
  /   (n + 1)!
 /__,         
n = 1         
$$\sum_{n=1}^{\infty} \frac{1}{\left(n + 1\right)!}$$
Sum(1/factorial(n + 1), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{1}{\left(n + 1\right)!}$$
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
$$a_{n} = \frac{1}{\left(n + 1\right)!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\left(n + 2\right)!}{\left(n + 1\right)!}}\right|$$
Let's take the limit
we find
False

False
The rate of convergence of the power series
The answer [src]
-2 + E
$$-2 + e$$
-2 + E
Numerical answer [src]
0.718281828459045235360287471353
0.718281828459045235360287471353
The graph
Sum of series 1/(n+1)!

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