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(1-cosn)/n^4
Sum of series/ (1-cosn)/n^4

Sum of series (1-cosn)/n^4


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The solution

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  oo            
____            
\   `           
 \    1 - cos(n)
  \   ----------
  /        4    
 /        n     
/___,           
n = 1
$$\sum_{n=1}^{\infty} \frac{1 - \cos{\left(n \right)}}{n^{4}}$$ 
Sum((1 - cos(n))/n^4, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{1 - \cos{\left(n \right)}}{n^{4}}$$
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
$$a_{n} = \frac{1 - \cos{\left(n \right)}}{n^{4}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right)^{4} \left|{\frac{\cos{\left(n \right)} - 1}{\cos{\left(n + 1 \right)} - 1}}\right|}{n^{4}}\right)$$
Let's take the limit
we find
True

False
The rate of convergence of the power series
The graph
Sum of series (1-cosn)/n^4

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