From proof wiki (Abel's Lemma):
Let $\left \langle {a} \right \rangle$ and $\left \langle {b} \right \rangle$ be sequences in an arbitrary ring $R$.
Let $\displaystyle A_n = \sum_{i \mathop = m}^n {a_i}$ be the partial sum of $\left \langle {a} \right \rangle$ from $m$ to $n$.
Then:$\displaystyle \sum_{i \mathop = m}^n a_i b_i = \sum_{i \mathop = m}^{n-1} A_i \left({b_i - b_{i+1}}\right) + A_n b_n$
What does $A_i$ signify here? From the definition given for $A_n$ I would think that $A_i = \sum_{i \mathop =m}^i a_i = a_i$. Where am I wrong?
The variable $i$ in the definition $$A_n = \sum_{i \mathop = m}^n {a_i}$$ is just a "dummy variable", and can be changed without changing the meaning: it's just a placeholder that represents all the indices between $m$ and $n$ that you're plugging in. In particular, note that the sum doesn't make sense if you replace $n$ with $i$, since $i$ is being used simultaneously as a dummy variable inside the sum and as a parameter outside it. To make sense of the sum $A_i$, then, you should change the internal variable $i$ to something else like $j$: $$A_i = \sum_{j \mathop = m}^i {a_j}$$