# Ngô Quốc Anh

## August 15, 2009

### Summation by parts (Abel sum formula)

Suppose $\{f_k\}$ and $\{g_k\}$ are two sequences. Then,

$\displaystyle\sum_{k=m}^n f_k(g_{k+1}-g_k) = \left[f_{n+1}g_{n+1} - f_m g_m\right] - \sum_{k=m}^n g_{k+1}(f_{k+1}- f_k)$.

Using the forward difference operator $\Delta$, it can be stated more succinctly as

$\displaystyle\sum_{k=m}^n f_k\Delta g_k = \left[f_{n+1} g_{n+1} - f_m g_m\right] - \sum_{k=m}^n g_{k+1}\Delta f_k$,

Note that summation by parts is an analogue to the integration by parts formula,

$\displaystyle\int f\,dg = f g - \int g df$.

We also have the following identity

$\displaystyle\sum_{k=m}^n f_k(g_k-g_{k-1}) = \left[f_{n+1}g_n - f_m g_{m-1}\right] - \sum_{k=m}^n g_k(f_{k+1}- f_k)$.

Using the backward difference operator $\Delta$, it can be stated more succinctly as

$\displaystyle\sum_{k=m}^n f_k\Delta g_{k-1} = \left[f_{n+1}g_n - f_m g_{m-1}\right] - \sum_{k=m}^n g_k \Delta f_k$.