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  • Sum of series:
  • 0.5^n 0.5^n
  • factorial(6*n)/4^n factorial(6*n)/4^n
  • 56724.3*10^-1 56724.3*10^-1
  • 12/(4n^2+12n+5) 12/(4n^2+12n+5)

  • Identical expressions

  • (arctg(one /n)^n)*x^n
  • (arctg(1 divide by n) to the power of n) multiply by x to the power of n
  • (arctg(one divide by n) to the power of n) multiply by x to the power of n
  • (arctg(1/n)n)*xn
  • arctg1/nn*xn
  • (arctg(1/n)^n)x^n
  • (arctg(1/n)n)xn
  • arctg1/nnxn
  • arctg1/n^nx^n
  • (arctg(1 divide by n)^n)*x^n
Sum of series/ (arctg(1/n)^n)*x^n

Sum of series (arctg(1/n)^n)*x^n


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The solution

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

    Examples of finding the sum of a series

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