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.

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