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1/((2^x)*x!)
Sum of series/ 1/((2^x)*x!)

Sum of series 1/((2^x)*x!)


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The solution

You have entered [src] 
  oo       
____       
\   `      
 \      1  
  \   -----
  /    x   
 /    2 *x!
/___,      
x = 1
$$\sum_{x=1}^{\infty} \frac{1}{2^{x} x!}$$ 
Sum(1/(2^x*factorial(x)), (x, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{1}{2^{x} x!}$$
It is a series of species
$$a_{x} \left(c x - x_{0}\right)^{d x}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{x \to \infty} \left|{\frac{a_{x}}{a_{x + 1}}}\right|}{c}$$
In this case
$$a_{x} = \frac{1}{x!}$$
and
$$x_{0} = -2$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-2 + \lim_{x \to \infty} \left|{\frac{\left(x + 1\right)!}{x!}}\right|\right)$$
Let's take the limit
we find
$$\frac{1}{R} = \infty$$
$$R = 0$$
The rate of convergence of the power series
The answer [src] 
      1/2
-1 + e
$$-1 + e^{\frac{1}{2}}$$ 
-1 + exp(1/2)
Numerical answer [src] 
0.648721270700128146848650787814
0.648721270700128146848650787814
The graph
Sum of series 1/((2^x)*x!)

    Examples of finding the sum of a series

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