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cos^2n/3^n

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  • Identical expressions

  • cos^2n/ three ^n
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  • co sinus of e of squared n divide by three to the power of n
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Sum of series/ cos^2n/3^n

Sum of series cos^2n/3^n


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The solution

You have entered [src] 
  oo         
____         
\   `        
 \       2   
  \   cos (n)
   )  -------
  /       n  
 /       3   
/___,        
n = 1
$$\sum_{n=1}^{\infty} \frac{\cos^{2}{\left(n \right)}}{3^{n}}$$ 
Sum(cos(n)^2/3^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\cos^{2}{\left(n \right)}}{3^{n}}$$
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
$$a_{n} = \cos^{2}{\left(n \right)}$$
and
$$x_{0} = -3$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-3 + \lim_{n \to \infty}\left(\cos^{2}{\left(n \right)} \left|{\frac{1}{\cos^{2}{\left(n + 1 \right)}}}\right|\right)\right)$$
Let's take the limit
we find
$$\frac{1}{R} = \tilde{\infty} \left(-3 + \lim_{n \to \infty}\left(\cos^{2}{\left(n \right)} \left|{\frac{1}{\cos^{2}{\left(n + 1 \right)}}}\right|\right)\right)$$
$$R = 0 \left(-3 + \lim_{n \to \infty}\left(\cos^{2}{\left(n \right)} \left|{\frac{1}{\cos^{2}{\left(n + 1 \right)}}}\right|\right)\right)^{-1}$$
The rate of convergence of the power series
The answer [src] 
  oo             
 ___             
 \  `            
  \    -n    2   
  /   3  *cos (n)
 /__,            
n = 1
$$\sum_{n=1}^{\infty} 3^{- n} \cos^{2}{\left(n \right)}$$ 
Sum(3^(-n)*cos(n)^2, (n, 1, oo))
Numerical answer [src] 
0.160039932917179357621930867242
0.160039932917179357621930867242
The graph
Sum of series cos^2n/3^n

    Examples of finding the sum of a series

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