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  • Sum of series:
  • sin(nx)/n^2
  • 1/((3n-2)*(3n+1)) 1/((3n-2)*(3n+1))
  • sin(1/n) sin(1/n)
  • n*3^n+1 n*3^n+1

  • Identical expressions

  • sin(nx)/n^ two
  • sinus of (nx) divide by n squared
  • sinus of (nx) divide by n to the power of two
  • sin(nx)/n2
  • sinnx/n2
  • sin(nx)/n²
  • sin(nx)/n to the power of 2
  • sinnx/n^2
  • sin(nx) divide by n^2
Sum of series/ sin(nx)/n^2

Sum of series sin(nx)/n^2


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The solution

You have entered [src] 
  oo          
____          
\   `         
 \    sin(n*x)
  \   --------
  /       2   
 /       n    
/___,         
n = 1
$$\sum_{n=1}^{\infty} \frac{\sin{\left(n x \right)}}{n^{2}}$$ 
Sum(sin(n*x)/n^2, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\sin{\left(n x \right)}}{n^{2}}$$
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
$$a_{n} = \frac{\sin{\left(n x \right)}}{n^{2}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right)^{2} \left|{\frac{\sin{\left(n x \right)}}{\sin{\left(x \left(n + 1\right) \right)}}}\right|}{n^{2}}\right)$$
Let's take the limit
we find
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