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Sum of series/ 1-cos(pi/n)

Sum of series 1-cos(pi/n)

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  oo               
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 \  `              
  \   /       /pi\\
   )  |1 - cos|--||
  /   \       \n //
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n = 1

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

Sum(1 - cos(pi/n), (n, 1, oo))

The radius of convergence of the power series

Given number:

1 - \cos{\left(\frac{\pi}{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} = 1 - \cos{\left(\frac{\pi}{n} \right)}

and

x_{0} = 0

d = 0

c = 1

then

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

True

False

The rate of convergence of the power series

Numerical answer [src]

4.87071896189479740325580288923

4.87071896189479740325580288923

The graph