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  • Sum of series:
  • 1/((3n-2)(3n+1)) 1/((3n-2)(3n+1))
  • 3/(n(n+2)) 3/(n(n+2))
  • (n+1)/5^n (n+1)/5^n
  • 6/9n^2+12n-5 6/9n^2+12n-5

  • Identical expressions

  • arctg(x^ three)/(n*(n+ two)*(n+ three))
  • arctg(x cubed ) divide by (n multiply by (n plus 2) multiply by (n plus 3))
  • arctg(x to the power of three) divide by (n multiply by (n plus two) multiply by (n plus three))
  • arctg(x3)/(n*(n+2)*(n+3))
  • arctgx3/n*n+2*n+3
  • arctg(x³)/(n*(n+2)*(n+3))
  • arctg(x to the power of 3)/(n*(n+2)*(n+3))
  • arctg(x^3)/(n(n+2)(n+3))
  • arctg(x3)/(n(n+2)(n+3))
  • arctgx3/nn+2n+3
  • arctgx^3/nn+2n+3
  • arctg(x^3) divide by (n*(n+2)*(n+3))

  • Similar expressions

  • arctg(x^3)/(n*(n+2)*(n-3))
  • arctg(x^3)/(n*(n-2)*(n+3))
Sum of series/ arctg(x^3)/(n*(n+2)*(n+3))

Sum of series arctg(x^3)/(n*(n+2)*(n+3))


∑
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The solution

You have entered [src] 
  oo                   
____                   
\   `                  
 \             / 3\    
  \        atan\x /    
  /   -----------------
 /    n*(n + 2)*(n + 3)
/___,                  
n = 1
$$\sum_{n=1}^{\infty} \frac{\operatorname{atan}{\left(x^{3} \right)}}{n \left(n + 2\right) \left(n + 3\right)}$$ 
Sum(atan(x^3)/(((n*(n + 2))*(n + 3))), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\operatorname{atan}{\left(x^{3} \right)}}{n \left(n + 2\right) \left(n + 3\right)}$$
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
$$a_{n} = \frac{\operatorname{atan}{\left(x^{3} \right)}}{n \left(n + 2\right) \left(n + 3\right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right) \left(n + 4\right)}{n \left(n + 2\right)}\right)$$
Let's take the limit
we find
True

False
The answer [src] 
      / 3\
5*atan\x /
----------
    36
$$\frac{5 \operatorname{atan}{\left(x^{3} \right)}}{36}$$ 
5*atan(x^3)/36

    Examples of finding the sum of a series

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