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Sum of series/ 1/(x*ln(x)^2)

Sum of series 1/(x*ln(x)^2)


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The solution

You have entered [src] 
  oo           
____           
\   `          
 \        1    
  \   ---------
  /        2   
 /    x*log (x)
/___,          
n = 2
∑n=2∞1xlog⁡(x)2\sum_{n=2}^{\infty} \frac{1}{x \log{\left(x \right)}^{2}} 
Sum(1/(x*log(x)^2), (n, 2, oo))
The radius of convergence of the power series
Given number:
1xlog⁡(x)2\frac{1}{x \log{\left(x \right)}^{2}}
It is a series of species
an(cx−x0)dna_{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:
Rd=x0+lim⁡n→∞∣anan+1∣cR^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}
In this case
an=1xlog⁡(x)2a_{n} = \frac{1}{x \log{\left(x \right)}^{2}}
and
x0=0x_{0} = 0
,
d=0d = 0
,
c=1c = 1
then
1=lim⁡n→∞11 = \lim_{n \to \infty} 1
Let's take the limit
we find
True

False
The answer [src] 
    oo   
---------
     2   
x*log (x)
∞xlog⁡(x)2\frac{\infty}{x \log{\left(x \right)}^{2}} 
oo/(x*log(x)^2)

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