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Sum of series lnx^5(1-2n)



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The solution

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  oo                   
 ___                   
 \  `                  
  \      5             
  /   log (x)*(1 - 2*n)
 /__,                  
n = 1                  
$$\sum_{n=1}^{\infty} \left(1 - 2 n\right) \log{\left(x \right)}^{5}$$
Sum(log(x)^5*(1 - 2*n), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\left(1 - 2 n\right) \log{\left(x \right)}^{5}$$
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
$$a_{n} = \left(1 - 2 n\right) \log{\left(x \right)}^{5}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left|{2 n - 1}\right|}{2 n + 1}\right)$$
Let's take the limit
we find
True

False
The answer [src]
       5   
-oo*log (x)
$$- \infty \log{\left(x \right)}^{5}$$
-oo*log(x)^5

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