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1/(x*(1-1/x)^2)

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  • Similar expressions

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

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


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The solution

You have entered [src] 
  oo             
_____            
\    `           
 \         1     
  \    ----------
   \            2
   /     /    1\ 
  /    x*|1 - -| 
 /       \    x/ 
/____,           
x = 2
$$\sum_{x=2}^{\infty} \frac{1}{x \left(1 - \frac{1}{x}\right)^{2}}$$ 
Sum(1/(x*(1 - 1/x)^2), (x, 2, oo))
The radius of convergence of the power series
Given number:
$$\frac{1}{x \left(1 - \frac{1}{x}\right)^{2}}$$
It is a series of species
$$a_{x} \left(c x - x_{0}\right)^{d x}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{x \to \infty} \left|{\frac{a_{x}}{a_{x + 1}}}\right|}{c}$$
In this case
$$a_{x} = \frac{1}{x \left(1 - \frac{1}{x}\right)^{2}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{x \to \infty}\left(\frac{\left(1 - \frac{1}{x + 1}\right)^{2} \left(x + 1\right) \left|{\frac{1}{\left(1 - \frac{1}{x}\right)^{2}}}\right|}{x}\right)$$
Let's take the limit
we find
True

False
The rate of convergence of the power series
The answer [src] 
oo
$$\infty$$ 
oo
Numerical answer
The series diverges
The graph
Sum of series 1/(x*(1-1/x)^2)

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