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(cos*1/n)/n^2
Sum of series/ (cos*1/n)/n^2

Sum of series (cos*1/n)/n^2


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The solution

You have entered [src] 
  oo           
_____          
\    `         
 \     /cos(1)\
  \    |------|
   \   \  n   /
   /   --------
  /        2   
 /        n    
/____,         
n = 1
$$\sum_{n=1}^{\infty} \frac{\frac{1}{n} \cos{\left(1 \right)}}{n^{2}}$$ 
Sum((cos(1)/n)/n^2, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\frac{1}{n} \cos{\left(1 \right)}}{n^{2}}$$
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
$$a_{n} = \frac{\cos{\left(1 \right)}}{n^{3}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right)^{3}}{n^{3}}\right)$$
Let's take the limit
we find
True

False
The rate of convergence of the power series
The answer [src] 
cos(1)*zeta(3)
$$\cos{\left(1 \right)} \zeta\left(3\right)$$ 
cos(1)*zeta(3)
Numerical answer [src] 
0.649474116561843915475121580529
0.649474116561843915475121580529
The graph
Sum of series (cos*1/n)/n^2

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