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

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

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

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

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

The radius of convergence of the power series

Given number:

\frac{1}{n^{n - 1}}

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} = n^{1 - n}

and

x_{0} = 0

d = 0

c = 1

then

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

False

False

The rate of convergence of the power series

The answer [src]

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

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

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

Numerical answer [src]

1.62847371290158444705588914326

1.62847371290158444705588914326

The graph