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1/(xln(x))
Sum of series/ 1/(xln(x))

Sum of series 1/(xln(x))


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The solution

You have entered [src] 
  oo          
 ___          
 \  `         
  \      1    
   )  --------
  /   x*log(x)
 /__,         
x = 1
$$\sum_{x=1}^{\infty} \frac{1}{x \log{\left(x \right)}}$$ 
Sum(1/(x*log(x)), (x, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{1}{x \log{\left(x \right)}}$$
It is a series of species
$$a_{x} \left(c x - x_{0}\right)^{d x}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{x \to \infty} \left|{\frac{a_{x}}{a_{x + 1}}}\right|}{c}$$
In this case
$$a_{x} = \frac{1}{x \log{\left(x \right)}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{x \to \infty}\left(\frac{\left(x + 1\right) \log{\left(x + 1 \right)} \left|{\frac{1}{\log{\left(x \right)}}}\right|}{x}\right)$$
Let's take the limit
we find
True

False
The rate of convergence of the power series
Numerical answer [src] 
Sum(1/(x*log(x)), (x, 1, oo))
Sum(1/(x*log(x)), (x, 1, oo))
The graph
Sum of series 1/(xln(x))

    Examples of finding the sum of a series

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