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ln(5)^n/n!

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



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The solution

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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
$$R = \tilde{\infty} \left(- \log{\left(5 \right)} + \lim_{n \to \infty} \left|{\frac{\left(n + 1\right)!}{n!}}\right|\right)$$
Let's take the limit
we find
$$R = \infty$$
The rate of convergence of the power series
The answer [src]
4
$$4$$
4
Numerical answer [src]
4.00000000000000000000000000000
4.00000000000000000000000000000
The graph
Sum of series ln(5)^n/n!

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