sigma(n=3 on bottom, n=23 on top)n/(n+4)
You are showing a misunderstanding of the notation.
Frequently n is used to show the final value of the index. The index is frequently shown as i. So the summation shows the initial value of the index on the bottom of the sigma, with the final value on top of the sigma.
So the most intuitive and conventional formulation would be
\(\displaystyle \displaystyle \sum_{i = 3}^6\dfrac{i}{i + 4}.\)
When i = 3, then the fraction is 3/(3 + 4) = 3/7. When i = 4, the fraction is 4/(2 + 4) = 4/8. And so on.
Some people abbreviate it as follows
\(\displaystyle \displaystyle \sum_3^6\dfrac{i}{i+4}.\)
But you can start the index anywhere. For example,
\(\displaystyle \displaystyle \sum_{i=1}^4\dfrac{i + 2}{i + 6}.\)
When i = 1, the fraction is (1 + 2)/(1 + 6) = 3/7. When i = 2, the fraction is (2 + 2)/(2 + 6) = 4/8. And so on.
An advantage of this approach is that it tells you very clearly that there are four summands in the sum.
And as Mark has pointed out, there are many other ways to express the same sum in sigma notation. It is very flexible.
