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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

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  /   x*(x - 1)
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x = 1
∑x=1∞1x(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:
1x(x−1)\frac{1}{x \left(x - 1\right)}
It is a series of species
ax(cx−x0)dxa_{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:
Rd=x0+lim⁡x→∞∣axax+1∣cR^{d} = \frac{x_{0} + \lim_{x \to \infty} \left|{\frac{a_{x}}{a_{x + 1}}}\right|}{c}
In this case
ax=1x(x−1)a_{x} = \frac{1}{x \left(x - 1\right)}
and
x0=0x_{0} = 0
,
d=0d = 0
,
c=1c = 1
then
1=lim⁡x→∞((x+1)∣1x−1∣)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
-0.010-0.008-0.006-0.004-0.0020.0100.0000.0020.0040.0060.0080.00
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))

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