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Sum of series sqrtx^x/x!



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The solution

You have entered [src]
  oo        
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\   `       
 \         x
  \     ___ 
   )  \/ x  
  /   ------
 /      x!  
/___,       
n = 1       
$$\sum_{n=1}^{\infty} \frac{\left(\sqrt{x}\right)^{x}}{x!}$$
Sum((sqrt(x))^x/factorial(x), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\left(\sqrt{x}\right)^{x}}{x!}$$
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
$$a_{n} = \frac{x^{\frac{x}{2}}}{x!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} 1$$
Let's take the limit
we find
True

False
The answer [src]
    x
    -
    2
oo*x 
-----
  x! 
$$\frac{\infty x^{\frac{x}{2}}}{x!}$$
oo*x^(x/2)/factorial(x)

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