Sum of series 1/nln(n)

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  oo        
 ___        
 \  `       
  \   log(n)
   )  ------
  /     n   
 /__,       
n = 2

\sum_{n=2}^{\infty} \frac{\log{\left(n \right)}}{n}

Sum(log(n)/n, (n, 2, oo))

The radius of convergence of the power series

Given number:

\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 \right)}}{n}

and

x_{0} = 0

d = 0

c = 1

then

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

True

False

The rate of convergence of the power series

Numerical answer

The series diverges

The graph