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  • Sum of series:
  • x^n
  • 1/n*(n+1)*(n+2) 1/n*(n+1)*(n+2)
  • 1/((2n+5)(2n+7)) 1/((2n+5)(2n+7))
  • 6/((3n-1)(3n+5)) 6/((3n-1)(3n+5))

  • Identical expressions

  • n^(one / three)*((x+ one)/(x))^ three
  • n to the power of (1 divide by 3) multiply by ((x plus 1) divide by (x)) cubed
  • n to the power of (one divide by three) multiply by ((x plus one) divide by (x)) to the power of three
  • n(1/3)*((x+1)/(x))3
  • n1/3*x+1/x3
  • n^(1/3)*((x+1)/(x))³
  • n to the power of (1/3)*((x+1)/(x)) to the power of 3
  • n^(1/3)((x+1)/(x))^3
  • n(1/3)((x+1)/(x))3
  • n1/3x+1/x3
  • n^1/3x+1/x^3
  • n^(1 divide by 3)*((x+1) divide by (x))^3

  • Similar expressions

  • n^(1/3)*((x-1)/(x))^3
Sum of series/ n^(1/3)*((x+1)/(x))^3

Sum of series n^(1/3)*((x+1)/(x))^3


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The solution

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

False
The answer [src] 
  oo                
____                
\   `               
 \    3 ___        3
  \   \/ n *(1 + x) 
   )  --------------
  /          3      
 /          x       
/___,               
n = 1
$$\sum_{n=1}^{\infty} \frac{\sqrt[3]{n} \left(x + 1\right)^{3}}{x^{3}}$$ 
Sum(n^(1/3)*(1 + x)^3/x^3, (n, 1, oo))

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