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Sum of series/ ln(5)^n/n!

Sum of series ln(5)^n/n!

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

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

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

The radius of convergence of the power series

Given number:

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

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!}

and

x_{0} = - \log{\left(5 \right)}

d = 1

c = 0

then

R = \tilde{\infty} \left(- \log{\left(5 \right)} + \lim_{n \to \infty} \left|{\frac{\left(n + 1\right)!}{n!}}\right|\right)

R = \infty

The rate of convergence of the power series

The answer [src]

4

Numerical answer [src]

4.00000000000000000000000000000

4.00000000000000000000000000000

The graph