I'm tracing through solution for a question I was working on. I don't quite understand how they got to the line which I marked with an arrow (apologies for using an image, I don't have much time left and didn't wanna have to look up how to do sigma notation in LaTeX here)
Text from the image:
The given sum is $\begin{align*} &7\sum_{k=2}^n \frac1{k-1}-3\sum_{k=2}^n\frac1k-4\sum_{k=2}^n\frac1{k+1}\\ =&7\sum_{k=1}^{n-1} \frac1k-3\sum_{k=2}^n\frac1k-4\sum_{k=3}^{n+1}\frac1k\\ \color{red}{\longrightarrow}\overset{?}=&7\left(1+\frac12\right)-3\left(\frac12+\frac1n\right)-4\left(\frac1n+\frac1{n+1}\right)\\ &=9-\frac7n-\frac4{n+1} \end{align*}$
So I understand how they got the second line, which is just basically taking the coefficients out and using the change of summation index rule.
What I don't understand is the line marked with the red arrow.
I can see that $1$ comes from $1/1$ when $k=1$ and $1/2$ probably comes from $k=2$ but the summation is from $k=1$ to $k=n-1$
How do we know to use $k=2$ and to stop at $k=2$ for that first term?