Mister Exam

Sum of series lnx



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The solution

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

False
The rate of convergence of the power series
Numerical answer
The series diverges
The graph
Sum of series lnx

    Examples of finding the sum of a series