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

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


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The solution

You have entered [src] 
  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
$$1 = \lim_{n \to \infty}\left(n \left|{\frac{\cos{\left(\pi n \right)}}{\left(n - 1\right) \cos{\left(\pi \left(n + 1\right) \right)}}}\right|\right)$$
Let's take the limit
we find
$$1 = \lim_{n \to \infty}\left(n \left|{\frac{\cos{\left(\pi n \right)}}{\left(n - 1\right) \cos{\left(\pi \left(n + 1\right) \right)}}}\right|\right)$$
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
Sum of series cos(pi*n)/(n-1)

    Examples of finding the sum of a series

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