I have a question about sigma notation. Here was what I was working on and its solution.

With each of these terms the denominator is 3 more than the numerator.

You could choose any letter (pronumeral) but they have chose the letter k to be a numerator.

If the numerator is k then the denominator would be k+3

So one way to describe each of these terms is     $$\frac{k}{k+3}$$    so this can be used as the general term.

Now the first term is    $${\frac{{\mathtt{4}}}{{\mathtt{7}}}}$$      so for the first term k=4

The second term is      $${\frac{{\mathtt{5}}}{{\mathtt{8}}}}$$       so for the first term k=5

etc

The last  term is         $${\frac{{\mathtt{10}}}{{\mathtt{13}}}}$$       so for the first term k=10

so it is the sum of all those terms from k=4 to k=10

$$\sum\limits_{\;k=4}^{10}}\;\frac{k}{k+3}$$

Do you understand any better?

This could have been expressed in many ways

For instance

$$\\\sum\limits_{\;k=7}^{13}}\;\frac{k-3}{k}\\\\\\
\sum\limits_{\;a=7}^{13}}\;\frac{a-3}{a}\\\\\\
\sum\limits_{\;g=1}^{7}}\;\frac{g+3}{g+6}\\\\\\$$

these all describe exactly the same sum