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Sum of series sqrtx^x/x!

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  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

True

False

The answer [src]

    x
    -
    2
oo*x 
-----
  x!

\frac{\infty x^{\frac{x}{2}}}{x!}

oo*x^(x/2)/factorial(x)