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Sum of series/ x^n/(ln(n))^n

Sum of series x^n/(ln(n))^n


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The solution

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

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