Mathematics for Computer Science

13.6. Double Trouble 527

we are summing, they form a triangle:

j

1 2 3 4 5 ::: n

k 1 1

2 1 1=2

3 1 1=2 1=3

4 1 1=2 1=3 1=4

:::

n 1 1=2 ::: 1=n

The summation in equation 13.26 is summing each row and then adding the row

sums. Instead, we can sum the columns and then add the column sums. Inspecting

the table we see that this double sum can be written as

Xn

kD 1

HkD

Xn

kD 1

Xk

jD 1

1

j

D

Xn

jD 1

Xn

kDj

1

j

D

Xn

jD 1

1

j

Xn

kDj

1

D

Xn

jD 1

1

j

.njC1/

D

Xn

jD 1

nC 1

j

Xn

jD 1

j

j

D.nC1/

Xn

jD 1

1

j

Xn

jD 1

1

D.nC1/Hnn: (13.27)