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

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


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The solution

You have entered [src] 
  oo                
 ___                
 \  `               
  \     log(x) - 1  
   )  --------------
  /   log(x + 1) - 1
 /__,               
n = 1
$$\sum_{n=1}^{\infty} \frac{\log{\left(x \right)} - 1}{\log{\left(x + 1 \right)} - 1}$$ 
Sum((log(x) - 1)/(log(x + 1) - 1), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\log{\left(x \right)} - 1}{\log{\left(x + 1 \right)} - 1}$$
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
$$a_{n} = \frac{\log{\left(x \right)} - 1}{\log{\left(x + 1 \right)} - 1}$$
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*(-1 + log(x))
----------------
-1 + log(1 + x)
$$\frac{\infty \left(\log{\left(x \right)} - 1\right)}{\log{\left(x + 1 \right)} - 1}$$ 
oo*(-1 + log(x))/(-1 + log(1 + x))

    Examples of finding the sum of a series

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