Summation by parts

Remarks

The product rule for differences is $\Delta(f g)_k = f_k \Delta g_k + g_{k+1} \Delta f_k$, which one checks by expanding. Summing it from $k = m$ to $n$, the left side telescopes to $f_{n+1} g_{n+1} - f_m g_m$, and this gives the formula. A common form, often called the Abel transformation, uses partial sums $A_k = a_1 + \dots + a_k$: $$\sum_{k=1}^{n} a_k b_k = A_n b_n - \sum_{k=1}^{n-1} A_k (b_{k+1} - b_k).$$ Example: with $f_k = k$ and $g_k = 2^k$, $\Delta g_k = 2^k$ and the formula gives $\sum_{k=0}^{n} k\, 2^k = (n+1) 2^{n+1} - \sum_{k=0}^{n} 2^{k+1} = (n - 1) 2^{n+1} + 2$. For $n = 2$ both sides equal $10$.

Going deeper

For a student

The Abel form is the key step in Dirichlet's test and Abel's test for the convergence of series $\sum a_k b_k$. In Dirichlet's test, bounded partial sums $A_k$ and monotone $b_k$ tending to $0$ make the right side converge. Its continuous-variable version is Abel's summation formula, used in analytic number theory. In the calculus of finite differences it is proved from the fundamental theorem of the calculus of finite differences applied to the product $f g$.