# How to find common ratio of a geometric series.

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.

• This fails because the sequence is not geometric. The very definition of geometric series gives immediately that (for nonzero series) the ratio of two successive elements is the same. Oct 28, 2014 at 13:23
• Harmonic Progressions don't have common ratio
– user171358
Oct 28, 2014 at 13:23
• The above series is not geometric. Oct 28, 2014 at 13:24
• You were right in your method to find $r$, by the way. But as others have pointed out, this isn't a geometric series. And you actually proved it because the $r$ you found from the first to second term isn't the same $r$ from the second to third term. Oct 28, 2014 at 13:25
• This is a perfectly good question: the OP knows what he is asking, he has tried to solve the problem on his own and has posted his work. There is absolutely no reason to downvote this question!
– 5xum
Oct 28, 2014 at 13:34