Mister Exam

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

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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)

True

False

The answer [src]

       5   
-oo*log (x)

- \infty \log{\left(x \right)}^{5}

-oo*log(x)^5