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Sum of series/ sinx/(x^p+sinx)

Sum of series sinx/(x^p+sinx)


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The solution

You have entered [src] 
  oo             
____             
\   `            
 \       sin(x)  
  \   -----------
  /    p         
 /    x  + sin(x)
/___,            
n = 0
$$\sum_{n=0}^{\infty} \frac{\sin{\left(x \right)}}{x^{p} + \sin{\left(x \right)}}$$ 
Sum(sin(x)/(x^p + sin(x)), (n, 0, oo))
The radius of convergence of the power series
Given number:
$$\frac{\sin{\left(x \right)}}{x^{p} + \sin{\left(x \right)}}$$
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
$$a_{n} = \frac{\sin{\left(x \right)}}{x^{p} + \sin{\left(x \right)}}$$
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] 
 oo*sin(x) 
-----------
 p         
x  + sin(x)
$$\frac{\infty \sin{\left(x \right)}}{x^{p} + \sin{\left(x \right)}}$$ 
oo*sin(x)/(x^p + sin(x))

    Examples of finding the sum of a series

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