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

Sum of series cos(pi*n)/n


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The solution

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  oo           
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 \  `          
  \   cos(pi*n)
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  /       n    
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n = 1
$$\sum_{n=1}^{\infty} \frac{\cos{\left(\pi n \right)}}{n}$$ 
Sum(cos(pi*n)/n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\cos{\left(\pi n \right)}}{n}$$
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
$$a_{n} = \frac{\cos{\left(\pi n \right)}}{n}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right) \left|{\frac{\cos{\left(\pi n \right)}}{\cos{\left(\pi \left(n + 1\right) \right)}}}\right|}{n}\right)$$
Let's take the limit
we find
True

False
The rate of convergence of the power series
The graph
Sum of series cos(pi*n)/n

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