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arctan(n)/(n!)^2

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  • Sum of series:
  • 38 38
  • arctan(n)/(n!)^2 arctan(n)/(n!)^2
  • arctg(5/n)/n! arctg(5/n)/n!
  • ln((n^2+1)/(n^2+n+2)) ln((n^2+1)/(n^2+n+2))

  • Identical expressions

  • arctan(n)/(n!)^ two
  • arc tangent of (n) divide by (n!) squared
  • arc tangent of (n) divide by (n!) to the power of two
  • arctan(n)/(n!)2
  • arctann/n!2
  • arctan(n)/(n!)²
  • arctan(n)/(n!) to the power of 2
  • arctann/n!^2
  • arctan(n) divide by (n!)^2
Sum of series/ arctan(n)/(n!)^2

Sum of series arctan(n)/(n!)^2


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The solution

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

False
The rate of convergence of the power series
Numerical answer [src] 
1.09928096924132089068108027685
1.09928096924132089068108027685
The graph
Sum of series arctan(n)/(n!)^2

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