I have this series:
$$S_n = \sum_{k=1}^\infty \frac{5}{k} = 5+\frac{5}{2} + \frac{5}{3} + .... + \frac{5}{k}$$
I wonder if theres a method to find the $r$ for any geometric series, Because sometimes I can't "see" it.
I thought of something like that :
I need to find $r$ from $5$ to $\frac{5}{2}$ hence,
$$5r=\frac{5}{2} \rightarrow r = \frac{1}{2}$$
The problem is that it works only for $S_2$.
However If I calculate for $S_3$ using this method I get wrong answer.
$$S_3=5+\frac{1}{2}*5+(\frac{1}{2})^2*5 = 5+\frac{5}{2}+\frac{5}{4} \neq 5+\frac{5}{2}+\frac{5}{3} = S_3$$
I wonder if there is a method that could work on every geometric series.
Any help will be appreciated.