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Sum of series/ (lnx-1)/(ln(n+1)-1)

Sum of series (lnx-1)/(ln(n+1)-1)


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