Mister Exam

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

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  oo            
 ___            
 \  `           
  \      /n + 1\
   )  log|-----|
  /      \n + 2/
 /__,           
n = 4

\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{\left(\frac{n + 1}{n + 2} \right)}

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} = \log{\left(\frac{n + 1}{n + 2} \right)}

and

x_{0} = 0

d = 0

c = 1

then

1 = \lim_{n \to \infty}\left(\frac{\log{\left(\frac{n + 1}{n + 2} \right)}}{\log{\left(\frac{n + 2}{n + 3} \right)}}\right)

True

False

The rate of convergence of the power series

Numerical answer

The series diverges

The graph