Sum of series log5(n-i)

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  oo            
 ___            
 \  `           
  \   log(n - i)
   )  ----------
  /     log(5)  
 /__,           
i = 0

\sum_{i=0}^{\infty} \frac{\log{\left(- i + n \right)}}{\log{\left(5 \right)}}

Sum(log(n - i)/log(5), (i, 0, oo))

The radius of convergence of the power series

Given number:

\frac{\log{\left(- i + n \right)}}{\log{\left(5 \right)}}

It is a series of species

a_{i} \left(c x - x_{0}\right)^{d i}

- power series.
The radius of convergence of a power series can be calculated by the formula:

R^{d} = \frac{x_{0} + \lim_{i \to \infty} \left|{\frac{a_{i}}{a_{i + 1}}}\right|}{c}

In this case

a_{i} = \frac{\log{\left(- i + n \right)}}{\log{\left(5 \right)}}

and

x_{0} = 0

d = 0

c = 1

then

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

True

False