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Sum of series/ (x-arctgx)/x^2

Sum of series (x-arctgx)/x^2


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The solution

You have entered [src] 
  oo             
____             
\   `            
 \    x - acot(x)
  \   -----------
  /         2    
 /         x     
/___,            
n = 1
$$\sum_{n=1}^{\infty} \frac{x - \operatorname{acot}{\left(x \right)}}{x^{2}}$$ 
Sum((x - acot(x))/x^2, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{x - \operatorname{acot}{\left(x \right)}}{x^{2}}$$
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
$$a_{n} = \frac{x - \operatorname{acot}{\left(x \right)}}{x^{2}}$$
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*(x - acot(x))
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        2       
       x
$$\frac{\infty \left(x - \operatorname{acot}{\left(x \right)}\right)}{x^{2}}$$ 
oo*(x - acot(x))/x^2

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