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

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  • Sum of series:
  • 2^(n+1)/factorial(n+1) 2^(n+1)/factorial(n+1)
  • 1/5 1/5
  • (sin(pi/n^3))^2n (sin(pi/n^3))^2n
  • sin(n*x)

  • 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
∑n=1∞nsin⁡2(πn3)\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:
nsin⁡2(πn3)n \sin^{2}{\left(\frac{\pi}{n^{3}} \right)}
It is a series of species
an(cx−x0)dna_{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:
Rd=x0+lim⁡n→∞∣anan+1∣cR^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}
In this case
an=nsin⁡2(πn3)a_{n} = n \sin^{2}{\left(\frac{\pi}{n^{3}} \right)}
and
x0=0x_{0} = 0
,
d=0d = 0
,
c=1c = 1
then
1=lim⁡n→∞(nsin⁡2(πn3)∣1sin⁡2(π(n+1)3)∣n+1)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
1.07.01.52.02.53.03.54.04.55.05.56.06.50.00.5
Numerical answer [src] 
0.348738939060494682026741519783
0.348738939060494682026741519783
The graph
Sum of series (sin(pi/n^3))^2n

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