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log(3*n)/(n^2-n)
Sum of series/ log(3*n)/(n^2-n)

Sum of series log(3*n)/(n^2-n)


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The solution

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  oo          
____          
\   `         
 \    log(3*n)
  \   --------
  /     2     
 /     n  - n 
/___,         
n = 2
$$\sum_{n=2}^{\infty} \frac{\log{\left(3 n \right)}}{n^{2} - n}$$ 
Sum(log(3*n)/(n^2 - n), (n, 2, oo))
The radius of convergence of the power series
Given number:
$$\frac{\log{\left(3 n \right)}}{n^{2} - n}$$
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
$$a_{n} = \frac{\log{\left(3 n \right)}}{n^{2} - n}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left|{\frac{\left(n - \left(n + 1\right)^{2} + 1\right) \log{\left(3 n \right)}}{n^{2} - n}}\right|}{\log{\left(3 n + 3 \right)}}\right)$$
Let's take the limit
we find
True

False
The rate of convergence of the power series
The graph
Sum of series log(3*n)/(n^2-n)

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