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

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

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