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((ln(n+1))/(n+1))-(ln(n)/(n))
Sum of series/ ((ln(n+1))/(n+1))-(ln(n)/(n))

Sum of series ((ln(n+1))/(n+1))-(ln(n)/(n))


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The solution

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  oo                       
 ___                       
 \  `                      
  \   /log(n + 1)   log(n)\
   )  |---------- - ------|
  /   \  n + 1        n   /
 /__,                      
n = 5
$$\sum_{n=5}^{\infty} \left(\frac{\log{\left(n + 1 \right)}}{n + 1} - \frac{\log{\left(n \right)}}{n}\right)$$ 
Sum(log(n + 1)/(n + 1) - log(n)/n, (n, 5, oo))
The radius of convergence of the power series
Given number:
$$\frac{\log{\left(n + 1 \right)}}{n + 1} - \frac{\log{\left(n \right)}}{n}$$
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
$$a_{n} = \frac{\log{\left(n + 1 \right)}}{n + 1} - \frac{\log{\left(n \right)}}{n}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\frac{\log{\left(n + 1 \right)}}{n + 1} - \frac{\log{\left(n \right)}}{n}}{\frac{\log{\left(n + 2 \right)}}{n + 2} - \frac{\log{\left(n + 1 \right)}}{n + 1}}}\right|$$
Let's take the limit
we find
True

False
The rate of convergence of the power series
The answer [src] 
nan
$$\text{NaN}$$ 
nan
Numerical answer [src] 
-0.321887582486820074920151866645
-0.321887582486820074920151866645
The graph
Sum of series ((ln(n+1))/(n+1))-(ln(n)/(n))

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