Summation by Parts

n

8

g

digamma

harmonic number

∑

k

8

k

H

k



1

90

(k-1)k(2k-1)(5

6

k

-15

5

k

+5

4

k

+15

3

k

-

2

k

-9k-3)

H

k

-

(k-1)k(2800

7

k

+1225

6

k

-17075

5

k

-7625

4

k

+35047

3

k

+15202

2

k

-22878k-7128)

226800

The indefinite sum of a product

fg

can often be computed efficiently using summation by parts. For this technique to work effectively, the function

f

must have a simple expression for its indefinite sum while

g

must have a simple expression for its difference. Summation by parts provides a discrete analog for integration by parts that is used in ordinary infinitesimal calculus. This Demonstration considers the case when

f=

n

k

is a monomial in the summation variable

k

and

g

is either the sequence of harmonic numbers or the digamma function, both of which have simple differences.