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log((n+1)/(n+2))

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



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The solution

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  oo            
 ___            
 \  `           
  \      /n + 1\
   )  log|-----|
  /      \n + 2/
 /__,           
n = 4           
n=4log(n+1n+2)\sum_{n=4}^{\infty} \log{\left(\frac{n + 1}{n + 2} \right)}
Sum(log((n + 1)/(n + 2)), (n, 4, oo))
The radius of convergence of the power series
Given number:
log(n+1n+2)\log{\left(\frac{n + 1}{n + 2} \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(n+1n+2)a_{n} = \log{\left(\frac{n + 1}{n + 2} \right)}
and
x0=0x_{0} = 0
,
d=0d = 0
,
c=1c = 1
then
1=limn(log(n+1n+2)log(n+2n+3))1 = \lim_{n \to \infty}\left(\frac{\log{\left(\frac{n + 1}{n + 2} \right)}}{\log{\left(\frac{n + 2}{n + 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
Numerical answer
The series diverges
The graph
Sum of series log((n+1)/(n+2))

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