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(-1)^n*(2*n)!*(2/3)^(2*n+2)*(2*n+1)/(n!)^2

  • How to use it?

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

  • Identical expressions

  • (- one)^n*(two *n)!*(two / three)^(two *n+ two)*(two *n+ one)/(n!)^ two
  • ( minus 1) to the power of n multiply by (2 multiply by n)! multiply by (2 divide by 3) to the power of (2 multiply by n plus 2) multiply by (2 multiply by n plus 1) divide by (n!) squared
  • ( minus one) to the power of n multiply by (two multiply by n)! multiply by (two divide by three) to the power of (two multiply by n plus two) multiply by (two multiply by n plus one) divide by (n!) to the power of two
  • (-1)n*(2*n)!*(2/3)(2*n+2)*(2*n+1)/(n!)2
  • -1n*2*n!*2/32*n+2*2*n+1/n!2
  • (-1)^n*(2*n)!*(2/3)^(2*n+2)*(2*n+1)/(n!)²
  • (-1) to the power of n*(2*n)!*(2/3) to the power of (2*n+2)*(2*n+1)/(n!) to the power of 2
  • (-1)^n(2n)!(2/3)^(2n+2)(2n+1)/(n!)^2
  • (-1)n(2n)!(2/3)(2n+2)(2n+1)/(n!)2
  • -1n2n!2/32n+22n+1/n!2
  • -1^n2n!2/3^2n+22n+1/n!^2
  • (-1)^n*(2*n)!*(2 divide by 3)^(2*n+2)*(2*n+1) divide by (n!)^2

  • Similar expressions

  • (1)^n*(2*n)!*(2/3)^(2*n+2)*(2*n+1)/(n!)^2
  • (-1)^n*(2*n)!*(2/3)^(2*n-2)*(2*n+1)/(n!)^2
  • (-1)^n*(2*n)!*(2/3)^(2*n+2)*(2*n-1)/(n!)^2
Sum of series/ (-1)^n*(2*n)!*(2/3)^(2*n+2)*(2*n+1)/(n!)^2

Sum of series (-1)^n*(2*n)!*(2/3)^(2*n+2)*(2*n+1)/(n!)^2


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The solution

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

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