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1/(n!)^100
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  • Sum of series:
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  • (n+1)/5^n (n+1)/5^n
  • 6/9n^2+12n-5 6/9n^2+12n-5
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  • one /(n!)^ one hundred
  • 1 divide by (n!) to the power of 100
  • one divide by (n!) to the power of one hundred
  • 1/(n!)100
  • 1/n!100
  • 1/n!^100
  • 1 divide by (n!)^100

Sum of series 1/(n!)^100



=

The solution

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  oo       
____       
\   `      
 \      1  
  \   -----
  /     100
 /    n!   
/___,      
n = 1      
$$\sum_{n=1}^{\infty} \frac{1}{n!^{100}}$$
Sum(1/(factorial(n)^100), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{1}{n!^{100}}$$
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
$$a_{n} = \frac{1}{n!^{100}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\left|{\frac{1}{n!^{100}}}\right| \left(n + 1\right)!^{100}\right)$$
Let's take the limit
we find
False

False
The rate of convergence of the power series
The answer [src]
       _                                                                                                                                                                                                                                                                                                                
      |_   /                                                                                                                                                                                                                                                                                                        |  \
-1 +  |    |                                                                                                                                                                                                                                                                                                        | 1|
     0  99 \1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 |  /
$${{}_{0}F_{99}\left(\begin{matrix} \\ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 \end{matrix}\middle| {1} \right)} - 1$$
-1 + hyper((), (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1), 1)
Numerical answer [src]
1.00000000000000000000000000000
1.00000000000000000000000000000
The graph
Sum of series 1/(n!)^100

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