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

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


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The solution

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

False
The rate of convergence of the power series
The answer [src] 
zoo
$$\tilde{\infty}$$ 
±oo
Numerical answer [src] 
Sum(1/(x*(x - 1)), (x, 1, oo))
Sum(1/(x*(x - 1)), (x, 1, oo))
The graph
Sum of series 1/(x*(x-1))

    Examples of finding the sum of a series

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