Mister Exam

Lang:

Sum of series/ ln(n)/n^2

Sum of series ln(n)/n^2

You have entered [src]

  oo        
____        
\   `       
 \    log(n)
  \   ------
  /      2  
 /      n   
/___,       
n = 1

\sum_{n=1}^{\infty} \frac{\log{\left(n \right)}}{n^{2}}

Sum(log(n)/n^2, (n, 1, oo))

The radius of convergence of the power series

Given number:

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

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^{2}}

and

x_{0} = 0

d = 0

c = 1

then

1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right)^{2} \left|{\log{\left(n \right)}}\right|}{n^{2} \log{\left(n + 1 \right)}}\right)

True

False

The rate of convergence of the power series

Numerical answer [src]

0.937548254315843753702574094568

0.937548254315843753702574094568

The graph