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2^n/factorial(3*n)

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  • Sum of series:
  • (arcsin(1/n))^n (arcsin(1/n))^n
  • (2x-4)
  • 56724.3*10^-1 56724.3*10^-1
  • 2^n/factorial(3*n) 2^n/factorial(3*n)
  • Limit of the function:
  • 2^n/factorial(3*n)

  • Identical expressions

  • two ^n/factorial(three *n)
  • 2 to the power of n divide by factorial(3 multiply by n)
  • two to the power of n divide by factorial(three multiply by n)
  • 2n/factorial(3*n)
  • 2n/factorial3*n
  • 2^n/factorial(3n)
  • 2n/factorial(3n)
  • 2n/factorial3n
  • 2^n/factorial3n
  • 2^n divide by factorial(3*n)
Sum of series/ 2^n/factorial(3*n)

Sum of series 2^n/factorial(3*n)


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The solution

You have entered [src] 
  oo        
____        
\   `       
 \       n  
  \     2   
  /   ------
 /    (3*n)!
/___,       
n = 1
$$\sum_{n=1}^{\infty} \frac{2^{n}}{\left(3 n\right)!}$$ 
Sum(2^n/factorial(3*n), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{2^{n}}{\left(3 n\right)!}$$
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
$$a_{n} = \frac{1}{\left(3 n\right)!}$$
and
$$x_{0} = -2$$
,
$$d = 1$$
,
$$c = 0$$
then
$$R = \tilde{\infty} \left(-2 + \lim_{n \to \infty} \left|{\frac{\left(3 n + 3\right)!}{\left(3 n\right)!}}\right|\right)$$
Let's take the limit
we find
$$R = \infty$$
The rate of convergence of the power series
The answer [src] 
       _                   
      |_  /         |     \
-1 +  |   |         | 2/27|
     0  2 \1/3, 2/3 |     /
$${{}_{0}F_{2}\left(\begin{matrix} \\ \frac{1}{3}, \frac{2}{3} \end{matrix}\middle| {\frac{2}{27}} \right)} - 1$$ 
-1 + hyper((), (1/3, 2/3), 2/27)
Numerical answer [src] 
0.338910968171560180930401717345
0.338910968171560180930401717345
The graph
Sum of series 2^n/factorial(3*n)

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