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Sum of series/ cos(npi*2)/n^4

Sum of series cos(npi*2)/n^4

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  oo             
____             
\   `            
 \    cos(n*pi*2)
  \   -----------
  /         4    
 /         n     
/___,            
n = 1

\sum_{n=1}^{\infty} \frac{\cos{\left(2 \pi n \right)}}{n^{4}}

Sum(cos((n*pi)*2)/n^4, (n, 1, oo))

The radius of convergence of the power series

Given number:

\frac{\cos{\left(2 \pi n \right)}}{n^{4}}

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} = \frac{\cos{\left(2 \pi n \right)}}{n^{4}}

and

x_{0} = 0

d = 0

c = 1

then

1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right)^{4} \left|{\frac{\cos{\left(2 \pi n \right)}}{\cos{\left(\pi \left(2 n + 2\right) \right)}}}\right|}{n^{4}}\right)

True

False

The rate of convergence of the power series

The answer [src]

  oo             
____             
\   `            
 \    cos(2*pi*n)
  \   -----------
  /         4    
 /         n     
/___,            
n = 1

\sum_{n=1}^{\infty} \frac{\cos{\left(2 \pi n \right)}}{n^{4}}

Sum(cos(2*pi*n)/n^4, (n, 1, oo))

The graph