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arctg(n^3)/(n*(n+2)*(n+3))

  • How to use it?

  • Sum of series:
  • n^(1/n) n^(1/n)
  • 5^n 5^n
  • 2(1/(n^2+5n+6)) 2(1/(n^2+5n+6))
  • ((-1)^n)*((x^n)/(2n*n!))

  • Identical expressions

  • arctg(n^ three)/(n*(n+ two)*(n+ three))
  • arctg(n cubed ) divide by (n multiply by (n plus 2) multiply by (n plus 3))
  • arctg(n to the power of three) divide by (n multiply by (n plus two) multiply by (n plus three))
  • arctg(n3)/(n*(n+2)*(n+3))
  • arctgn3/n*n+2*n+3
  • arctg(n³)/(n*(n+2)*(n+3))
  • arctg(n to the power of 3)/(n*(n+2)*(n+3))
  • arctg(n^3)/(n(n+2)(n+3))
  • arctg(n3)/(n(n+2)(n+3))
  • arctgn3/nn+2n+3
  • arctgn^3/nn+2n+3
  • arctg(n^3) divide by (n*(n+2)*(n+3))

  • Similar expressions

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

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


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The solution

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

False
The rate of convergence of the power series
The answer [src] 
  oo                   
____                   
\   `                  
 \             / 3\    
  \        atan\n /    
  /   -----------------
 /    n*(2 + n)*(3 + n)
/___,                  
n = 1
$$\sum_{n=1}^{\infty} \frac{\operatorname{atan}{\left(n^{3} \right)}}{n \left(n + 2\right) \left(n + 3\right)}$$ 
Sum(atan(n^3)/(n*(2 + n)*(3 + n)), (n, 1, oo))
Numerical answer [src] 
0.149054087090755795491305522826
0.149054087090755795491305522826
The graph
Sum of series arctg(n^3)/(n*(n+2)*(n+3))

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