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Sum of series/ (arcsin(x)/x)^ctg(x)

Sum of series (arcsin(x)/x)^ctg(x)


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The solution

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

    Examples of finding the sum of a series

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