Mister Exam

Sum of series 1/nln(n)



=

The solution

You have entered [src]
  oo        
 ___        
 \  `       
  \   log(n)
   )  ------
  /     n   
 /__,       
n = 2       
n=2log(n)n\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:
log(n)n\frac{\log{\left(n \right)}}{n}
It is a series of species
an(cxx0)dna_{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:
Rd=x0+limnanan+1cR^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}
In this case
an=log(n)na_{n} = \frac{\log{\left(n \right)}}{n}
and
x0=0x_{0} = 0
,
d=0d = 0
,
c=1c = 1
then
1=limn((n+1)log(n)nlog(n+1))1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right) \left|{\log{\left(n \right)}}\right|}{n \log{\left(n + 1 \right)}}\right)
Let's take the limit
we find
True

False
The rate of convergence of the power series
2.08.02.53.03.54.04.55.05.56.06.57.07.504
Numerical answer
The series diverges
The graph
Sum of series 1/nln(n)

    Examples of finding the sum of a series