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1/(log(log(n))^(log(n)))

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  • Sum of series:
  • factorial(n)/(n^3+n^2+4) factorial(n)/(n^3+n^2+4)
  • ln(k/(k+1))
  • x/(x-1) x/(x-1)
  • sin(n*x)/5^n

  • Identical expressions

  • one /(log(log(n))^(log(n)))
  • 1 divide by ( logarithm of ( logarithm of (n)) to the power of ( logarithm of (n)))
  • one divide by ( logarithm of ( logarithm of (n)) to the power of ( logarithm of (n)))
  • 1/(log(log(n))(log(n)))
  • 1/loglognlogn
  • 1/loglogn^logn
  • 1 divide by (log(log(n))^(log(n)))
Sum of series/ 1/(log(log(n))^(log(n)))

Sum of series 1/(log(log(n))^(log(n)))


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The solution

You have entered [src] 
  oo                   
____                   
\   `                  
 \            1        
  \   -----------------
  /      log(n)        
 /    log      (log(n))
/___,                  
n = 3
$$\sum_{n=3}^{\infty} \frac{1}{\log{\left(\log{\left(n \right)} \right)}^{\log{\left(n \right)}}}$$ 
Sum(1/(log(log(n))^log(n)), (n, 3, oo))
The radius of convergence of the power series
Given number:
$$\frac{1}{\log{\left(\log{\left(n \right)} \right)}^{\log{\left(n \right)}}}$$
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
$$a_{n} = \log{\left(\log{\left(n \right)} \right)}^{- \log{\left(n \right)}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left|{\log{\left(\log{\left(n + 1 \right)} \right)}^{\log{\left(n + 1 \right)}}}\right|}{\left|{\log{\left(\log{\left(n \right)} \right)}^{\log{\left(n \right)}}}\right|}\right)$$
Let's take the limit
we find
$$1 = \lim_{n \to \infty}\left(\frac{\left|{\log{\left(\log{\left(n + 1 \right)} \right)}^{\log{\left(n + 1 \right)}}}\right|}{\left|{\log{\left(\log{\left(n \right)} \right)}^{\log{\left(n \right)}}}\right|}\right)$$
False
The rate of convergence of the power series
The answer [src] 
  oo                    
 ___                    
 \  `                   
  \      -log(n)        
  /   log       (log(n))
 /__,                   
n = 3
$$\sum_{n=3}^{\infty} \log{\left(\log{\left(n \right)} \right)}^{- \log{\left(n \right)}}$$ 
Sum(log(log(n))^(-log(n)), (n, 3, oo))
The graph
Sum of series 1/(log(log(n))^(log(n)))

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