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

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

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  oo           
 ___           
 \  `          
  \   cos(pi*n)
   )  ---------
  /     n - 1  
 /__,          
n = 1

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

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

The radius of convergence of the power series

Given number:

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

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(\pi n \right)}}{n - 1}

and

x_{0} = 0

d = 0

c = 1

then

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

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

False

The rate of convergence of the power series

Numerical answer [src]

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

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

The graph