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(8/n)(arctg(1/n^9))

  • How to use it?

  • Sum of series:
  • ln(1-1/n^2) ln(1-1/n^2)
  • cos*π*n cos*π*n
  • (1/(3n-2)(3n+1)) (1/(3n-2)(3n+1))
  • ln(n)/n^s

  • Identical expressions

  • (eight /n)(arctg(one /n^ nine))
  • (8 divide by n)(arctg(1 divide by n to the power of 9))
  • (eight divide by n)(arctg(one divide by n to the power of nine))
  • (8/n)(arctg(1/n9))
  • 8/narctg1/n9
  • (8/n)(arctg(1/n⁹))
  • 8/narctg1/n^9
  • (8 divide by n)(arctg(1 divide by n^9))
Sum of series/ (8/n)(arctg(1/n^9))

Sum of series (8/n)(arctg(1/n^9))


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The solution

You have entered [src] 
  oo            
____            
\   `           
 \    8     /1 \
  \   -*atan|--|
  /   n     | 9|
 /          \n /
/___,           
n = 1
$$\sum_{n=1}^{\infty} \frac{8}{n} \operatorname{atan}{\left(\frac{1}{n^{9}} \right)}$$ 
Sum((8/n)*atan(1/(n^9)), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{8}{n} \operatorname{atan}{\left(\frac{1}{n^{9}} \right)}$$
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
$$a_{n} = \frac{8 \operatorname{atan}{\left(\frac{1}{n^{9}} \right)}}{n}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right) \operatorname{atan}{\left(\frac{1}{n^{9}} \right)}}{n \operatorname{atan}{\left(\frac{1}{\left(n + 1\right)^{9}} \right)}}\right)$$
Let's take the limit
we find
True

False
The rate of convergence of the power series
The answer [src] 
  oo             
_____            
\    `           
 \           /1 \
  \    8*atan|--|
   \         | 9|
   /         \n /
  /    ----------
 /         n     
/____,           
n = 1
$$\sum_{n=1}^{\infty} \frac{8 \operatorname{atan}{\left(\frac{1}{n^{9}} \right)}}{n}$$ 
Sum(8*atan(n^(-9))/n, (n, 1, oo))
Numerical answer [src] 
6.29114189826792983082567291611
6.29114189826792983082567291611
The graph
Sum of series (8/n)(arctg(1/n^9))

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