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log((k+1)/(k+2))

Sum of series log((k+1)/(k+2))



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The solution

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  oo            
 ___            
 \  `           
  \      /k + 1\
   )  log|-----|
  /      \k + 2/
 /__,           
k = 4           
k=4log(k+1k+2)\sum_{k=4}^{\infty} \log{\left(\frac{k + 1}{k + 2} \right)}
Sum(log((k + 1)/(k + 2)), (k, 4, oo))
The radius of convergence of the power series
Given number:
log(k+1k+2)\log{\left(\frac{k + 1}{k + 2} \right)}
It is a series of species
ak(cxx0)dka_{k} \left(c x - x_{0}\right)^{d k}
- power series.
The radius of convergence of a power series can be calculated by the formula:
Rd=x0+limkakak+1cR^{d} = \frac{x_{0} + \lim_{k \to \infty} \left|{\frac{a_{k}}{a_{k + 1}}}\right|}{c}
In this case
ak=log(k+1k+2)a_{k} = \log{\left(\frac{k + 1}{k + 2} \right)}
and
x0=0x_{0} = 0
,
d=0d = 0
,
c=1c = 1
then
1=limk(log(k+1k+2)log(k+2k+3))1 = \lim_{k \to \infty}\left(\frac{\log{\left(\frac{k + 1}{k + 2} \right)}}{\log{\left(\frac{k + 2}{k + 3} \right)}}\right)
Let's take the limit
we find
True

False
The rate of convergence of the power series
4.04.55.05.56.06.57.07.58.08.59.09.510.0-1.00.0
The graph
Sum of series log((k+1)/(k+2))

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