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  • Sum of series:
  • 7+k 7+k
  • (-1)^n/2n-1 (-1)^n/2n-1
  • 2 2
  • ln(n+1)/(n+1) ln(n+1)/(n+1)

  • Identical expressions

  • lnx^ five (one -2n)
  • lnx to the power of 5(1 minus 2n)
  • lnx to the power of five (one minus 2n)
  • lnx5(1-2n)
  • lnx51-2n
  • lnx⁵(1-2n)
  • lnx^51-2n

  • Similar expressions

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

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


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The solution

You have entered [src] 
  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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