How to Prove Abel's Formula ?
Abel's Formula is –
Let $(a_n)$ and $(b_n)$ be sequences of real numbers. The Abel’s formula reads, for $p \geq 2$,
$\sum_{n=1}^{p} a_nb_n $ = $\sum_{k=1}^{p-1} (a_k – a_{k+1}) (\sum_{l=1}^{k} b_l)$ + $ a_p \sum_{l=1}^{p} b_l $
or can be written as
$\sum_{n=1}^{p} a_nb_n $ = $ (a_1 – a_2)b_1 + (a_2 – a_3)(b_1 + b_2)+(a_3 – a_4)(b_1 + b_2 + b_3) + ··· + (a_{p-1} – a_p)(b_1 + ··· + b_{p-1}) + a_p(b_1 + ··· + b_p)$
Best Answer
No induction is required to prove Abel's formula:
Set $\,B_n=\displaystyle\sum_{l=1}^{n}b_l\,$; for all $n>1$, we have $b_n=B_n-B_{n-1}$, and $b_1=B_1$. With these notations, we can write: \begin{align*} \sum_{n=1}^p a_nb_n &= a_1 B_1+\sum_{n=2}^p a_n(B_n-B_{n-1})=\sum_{n=1}^p a_n B_n -\sum_{k=1}^{p-1} a_{k+1} B_k\\ &= a_pB_p +\sum_{k=1}^{p-1}(a_k - a_{k+1}) B_k \end{align*}