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Sum of series/ x^n/(2n-1)!

Sum of series x^n/(2n-1)!


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The solution

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

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