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



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The solution

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

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