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

  • Identical expressions

  • sin^n*(pi/ three ^n)
  • sinus of to the power of n multiply by ( Pi divide by 3 to the power of n)
  • sinus of to the power of n multiply by ( Pi divide by three to the power of n)
  • sinn*(pi/3n)
  • sinn*pi/3n
  • sin^n(pi/3^n)
  • sinn(pi/3n)
  • sinnpi/3n
  • sin^npi/3^n
  • sin^n*(pi divide by 3^n)
Sum of series/ sin^n*(pi/3^n)

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


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The solution

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

False
The answer [src] 
  oo              
 ___              
 \  `             
  \      n/    -n\
  /   sin \pi*3  /
 /__,             
n = 1
$$\sum_{n=1}^{\infty} \sin^{n}{\left(3^{- n} \pi \right)}$$ 
Sum(sin(pi*3^(-n))^n, (n, 1, oo))
Numerical answer [src] 
0.984570092949291571101300677353
0.984570092949291571101300677353

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