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Sum of series/ 1/lnx

Sum of series 1/lnx


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The solution

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

False
The answer [src] 
  oo  
------
log(x)
$$\frac{\infty}{\log{\left(x \right)}}$$ 
oo/log(x)

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