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Sum of series/ sinx^2/x^2

Sum of series sinx^2/x^2


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The solution

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

    Examples of finding the sum of a series

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