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
.n jC1/
D
Xn
jD 1
nC 1
j