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Sum of series/ 1/x-1(x+1)

Sum of series 1/x-1(x+1)


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The solution

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

False
The answer [src] 
   /     1    \
oo*|-1 + - - x|
   \     x    /
$$\infty \left(- x - 1 + \frac{1}{x}\right)$$ 
oo*(-1 + 1/x - x)

    Examples of finding the sum of a series

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