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Sum of series ln(1/x)



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The solution

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  oo        
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   )  log|-|
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n = 1       
n=1log(1x)\sum_{n=1}^{\infty} \log{\left(\frac{1}{x} \right)}
Sum(log(1/x), (n, 1, oo))
The radius of convergence of the power series
Given number:
log(1x)\log{\left(\frac{1}{x} \right)}
It is a series of species
an(cxx0)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+limnanan+1cR^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}
In this case
an=log(1x)a_{n} = \log{\left(\frac{1}{x} \right)}
and
x0=0x_{0} = 0
,
d=0d = 0
,
c=1c = 1
then
1=limn11 = \lim_{n \to \infty} 1
Let's take the limit
we find
True

False
The answer [src]
      /1\
oo*log|-|
      \x/
log(1x)\infty \log{\left(\frac{1}{x} \right)}
oo*log(1/x)

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