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Sum of series/ 1/ln(5^n)

Sum of series 1/ln(5^n)

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

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

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

The radius of convergence of the power series

Given number:

\frac{1}{\log{\left(5^{n} \right)}}

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

and

x_{0} = 0

d = 0

c = 1

then

1 = \lim_{n \to \infty} \left|{\frac{\log{\left(5^{n + 1} \right)}}{\log{\left(5^{n} \right)}}}\right|

True

False

The rate of convergence of the power series

The answer [src]

oo

\infty

oo

Numerical answer

The series diverges

The graph