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((n^2)^1/3)*atan(1/n^2)

  • How to use it?

  • Sum of series:
  • sin(3n)/(7n)^(1/5) sin(3n)/(7n)^(1/5)
  • arctg1/(2n^2) arctg1/(2n^2)
  • cos(i*n)/2^n cos(i*n)/2^n
  • 1/(n*ln(n)*(ln(ln(n)))^2) 1/(n*ln(n)*(ln(ln(n)))^2)

  • Identical expressions

  • ((n^ two)^ one / three)*atan(one /n^ two)
  • ((n squared ) to the power of 1 divide by 3) multiply by arc tangent of gent of (1 divide by n squared )
  • ((n to the power of two) to the power of one divide by three) multiply by arc tangent of gent of (one divide by n to the power of two)
  • ((n2)1/3)*atan(1/n2)
  • n21/3*atan1/n2
  • ((n²)^1/3)*atan(1/n²)
  • ((n to the power of 2) to the power of 1/3)*atan(1/n to the power of 2)
  • ((n^2)^1/3)atan(1/n^2)
  • ((n2)1/3)atan(1/n2)
  • n21/3atan1/n2
  • n^2^1/3atan1/n^2
  • ((n^2)^1 divide by 3)*atan(1 divide by n^2)

  • Similar expressions

  • ((n^2)^1/3)*arctan(1/n^2)
Sum of series/ ((n^2)^1/3)*atan(1/n^2)

Sum of series ((n^2)^1/3)*atan(1/n^2)


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The solution

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

False
The rate of convergence of the power series
The answer [src] 
  oo               
____               
\   `              
 \     2/3     /1 \
  \   n   *atan|--|
  /            | 2|
 /             \n /
/___,              
n = 1
$$\sum_{n=1}^{\infty} n^{\frac{2}{3}} \operatorname{atan}{\left(\frac{1}{n^{2}} \right)}$$ 
Sum(n^(2/3)*atan(n^(-2)), (n, 1, oo))
Numerical answer [src] 
3.37710784789833859668350009753
3.37710784789833859668350009753
The graph
Sum of series ((n^2)^1/3)*atan(1/n^2)

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