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  • Sum of series:
  • (n-1)/2^(n+1) (n-1)/2^(n+1)
  • 6/(n^2-10n+24) 6/(n^2-10n+24)
  • a^n
  • 2^(2*n-1)/9^n 2^(2*n-1)/9^n

  • Identical expressions

  • (two x+ one)/(x^ two *(n+ one)^2)
  • (2x plus 1) divide by (x squared multiply by (n plus 1) squared )
  • (two x plus one) divide by (x to the power of two multiply by (n plus one) squared )
  • (2x+1)/(x2*(n+1)2)
  • 2x+1/x2*n+12
  • (2x+1)/(x²*(n+1)²)
  • (2x+1)/(x to the power of 2*(n+1) to the power of 2)
  • (2x+1)/(x^2(n+1)^2)
  • (2x+1)/(x2(n+1)2)
  • 2x+1/x2n+12
  • 2x+1/x^2n+1^2
  • (2x+1) divide by (x^2*(n+1)^2)

  • Similar expressions

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

Sum of series (2x+1)/(x^2*(n+1)^2)


∑
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The solution

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

False
The answer [src] 
         2            2
     2*pi         2*pi 
-4 + -----   -4 + -----
       3            3  
---------- + ----------
   2*x             2   
                4*x
$$\frac{-4 + \frac{2 \pi^{2}}{3}}{2 x} + \frac{-4 + \frac{2 \pi^{2}}{3}}{4 x^{2}}$$ 
(-4 + 2*pi^2/3)/(2*x) + (-4 + 2*pi^2/3)/(4*x^2)

    Examples of finding the sum of a series

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