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sin^2(1/4n)

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  • Sum of series:
  • sin(1/n) sin(1/n)
  • n*3^n+1 n*3^n+1
  • (2^(n+2))/(7^n*9^(n-1)) (2^(n+2))/(7^n*9^(n-1))
  • sin^2(1/4n) sin^2(1/4n)

  • Identical expressions

  • sin^ two (one /4n)
  • sinus of squared (1 divide by 4n)
  • sinus of to the power of two (one divide by 4n)
  • sin2(1/4n)
  • sin21/4n
  • sin²(1/4n)
  • sin to the power of 2(1/4n)
  • sin^21/4n
  • sin^2(1 divide by 4n)
Sum of series/ sin^2(1/4n)

Sum of series sin^2(1/4n)


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The solution

You have entered [src] 
  oo         
 ___         
 \  `        
  \      2/n\
   )  sin |-|
  /       \4/
 /__,        
n = 1
$$\sum_{n=1}^{\infty} \sin^{2}{\left(\frac{n}{4} \right)}$$ 
Sum(sin(n/4)^2, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\sin^{2}{\left(\frac{n}{4} \right)}$$
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
$$a_{n} = \sin^{2}{\left(\frac{n}{4} \right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\sin^{2}{\left(\frac{n}{4} \right)} \left|{\frac{1}{\sin^{2}{\left(\frac{n}{4} + \frac{1}{4} \right)}}}\right|\right)$$
Let's take the limit
we find
$$1 = \lim_{n \to \infty}\left(\sin^{2}{\left(\frac{n}{4} \right)} \left|{\frac{1}{\sin^{2}{\left(\frac{n}{4} + \frac{1}{4} \right)}}}\right|\right)$$
False
The rate of convergence of the power series
Numerical answer
The series diverges
The graph
Sum of series sin^2(1/4n)

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