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Sum of series/ (1/n)^(n-3)

Sum of series (1/n)^(n-3)

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  oo          
____          
\   `         
 \       n - 3
  \   /1\     
  /   |-|     
 /    \n/     
/___,         
n = 1

\sum_{n=1}^{\infty} \left(\frac{1}{n}\right)^{n - 3}

Sum((1/n)^(n - 3), (n, 1, oo))

The radius of convergence of the power series

Given number:

\left(\frac{1}{n}\right)^{n - 3}

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} = \left(\frac{1}{n}\right)^{n - 3}

and

x_{0} = 0

d = 0

c = 1

then

1 = \lim_{n \to \infty}\left(n^{3 - n} \left(n + 1\right)^{n - 2}\right)

False

False

The rate of convergence of the power series

The answer [src]

  oo        
 ___        
 \  `       
  \    3 - n
  /   n     
 /__,       
n = 1

\sum_{n=1}^{\infty} n^{3 - n}

Sum(n^(3 - n), (n, 1, oo))

Numerical answer [src]

4.29507862687843037097334808608

4.29507862687843037097334808608

The graph