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ln*sqrt((k+1)/k)
Sum of series/ ln*sqrt((k+1)/k)

Sum of series ln*sqrt((k+1)/k)


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The solution

You have entered [src] 
  oo                    
____                    
\   `                   
 \               _______
  \             / k + 1 
  /   log(k)*  /  ----- 
 /           \/     k   
/___,                   
k = 4
$$\sum_{k=4}^{\infty} \sqrt{\frac{k + 1}{k}} \log{\left(k \right)}$$ 
Sum(log(k)*sqrt((k + 1)/k), (k, 4, oo))
The radius of convergence of the power series
Given number:
$$\sqrt{\frac{k + 1}{k}} \log{\left(k \right)}$$
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
$$a_{k} = \sqrt{\frac{k + 1}{k}} \log{\left(k \right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{k \to \infty}\left(\frac{\left(k + 1\right) \left|{\log{\left(k \right)}}\right|}{\sqrt{k} \sqrt{k + 2} \log{\left(k + 1 \right)}}\right)$$
Let's take the limit
we find
True

False
The rate of convergence of the power series
The answer [src] 
  oo                  
____                  
\   `                 
 \      _______       
  \   \/ 1 + k *log(k)
   )  ----------------
  /          ___      
 /         \/ k       
/___,                 
k = 4
$$\sum_{k=4}^{\infty} \frac{\sqrt{k + 1} \log{\left(k \right)}}{\sqrt{k}}$$ 
Sum(sqrt(1 + k)*log(k)/sqrt(k), (k, 4, oo))
The graph
Sum of series ln*sqrt((k+1)/k)

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