Mister Exam

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Sum of series 1/(n*(log(n))^5)

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  oo           
____           
\   `          
 \        1    
  \   ---------
  /        5   
 /    n*log (n)
/___,          
n = 1

\sum_{n=1}^{\infty} \frac{1}{n \log{\left(n \right)}^{5}}

Sum(1/(n*log(n)^5), (n, 1, oo))

The radius of convergence of the power series

Given number:

\frac{1}{n \log{\left(n \right)}^{5}}

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{1}{n \log{\left(n \right)}^{5}}

and

x_{0} = 0

d = 0

c = 1

then

1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right) \log{\left(n + 1 \right)}^{5} \left|{\frac{1}{\log{\left(n \right)}^{5}}}\right|}{n}\right)

True

False

The rate of convergence of the power series

The graph