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1/(lnn)^2
Sum of series/ 1/(lnn)^2

Sum of series 1/(lnn)^2


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The solution

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

False
The rate of convergence of the power series
The answer [src] 
  oo         
____         
\   `        
 \       1   
  \   -------
  /      2   
 /    log (n)
/___,        
n = 1
$$\sum_{n=1}^{\infty} \frac{1}{\log{\left(n \right)}^{2}}$$ 
Sum(log(n)^(-2), (n, 1, oo))
Numerical answer [src] 
Sum(1/(log(n)^2), (n, 1, oo))
Sum(1/(log(n)^2), (n, 1, oo))
The graph
Sum of series 1/(lnn)^2

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