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(sin(pi/n^3))^2n

  • How to use it?

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

  • Identical expressions

  • (sin(pi/n^ three))^2n
  • ( sinus of ( Pi divide by n cubed )) squared n
  • ( sinus of ( Pi divide by n to the power of three)) squared n
  • (sin(pi/n3))2n
  • sinpi/n32n
  • (sin(pi/n³))²n
  • (sin(pi/n to the power of 3)) to the power of 2n
  • sinpi/n^3^2n
  • (sin(pi divide by n^3))^2n
Sum of series/ (sin(pi/n^3))^2n

Sum of series (sin(pi/n^3))^2n


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The solution

You have entered [src] 
  oo            
____            
\   `           
 \       2/pi\  
  \   sin |--|*n
  /       | 3|  
 /        \n /  
/___,           
n = 1
$$\sum_{n=1}^{\infty} n \sin^{2}{\left(\frac{\pi}{n^{3}} \right)}$$ 
Sum(sin(pi/n^3)^2*n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$n \sin^{2}{\left(\frac{\pi}{n^{3}} \right)}$$
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
$$a_{n} = n \sin^{2}{\left(\frac{\pi}{n^{3}} \right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{n \sin^{2}{\left(\frac{\pi}{n^{3}} \right)} \left|{\frac{1}{\sin^{2}{\left(\frac{\pi}{\left(n + 1\right)^{3}} \right)}}}\right|}{n + 1}\right)$$
Let's take the limit
we find
True

False
The rate of convergence of the power series
Numerical answer [src] 
0.348738939060494682026741519783
0.348738939060494682026741519783
The graph
Sum of series (sin(pi/n^3))^2n

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