Mathematics for Computer Science
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“mcs” — 2017/3/3 — 11:21 — page 571 — #579<br />
14.6. Double Trouble 571<br />
we are summing, they <strong>for</strong>m a triangle:<br />
j<br />
1 2 3 4 5 : : : n<br />
k 1 1<br />
2 1 1=2<br />
3 1 1=2 1=3<br />
4 1 1=2 1=3 1=4<br />
: : :<br />
n 1 1=2 : : : 1=n<br />
The summation in equation 14.26 is summing each row and then adding the row<br />
sums. Instead, we can sum the columns and then add the column sums. Inspecting<br />
the table we see that this double sum can be written as<br />
nX<br />
H k D<br />
kD1<br />
D<br />
D<br />
D<br />
D<br />
nX<br />
kX<br />
kD1 j D1<br />
nX<br />
nX<br />
j D1 kDj<br />
nX<br />
j D1<br />
nX<br />
j D1<br />
nX<br />
j D1<br />
1<br />
j<br />
1<br />
j<br />
1<br />
j<br />
nX<br />
1<br />
kDj<br />
1<br />
.n j C 1/<br />
j<br />
n C 1<br />
j<br />
D .n C 1/<br />
nX<br />
j D1<br />
1<br />
j<br />
nX<br />
j D1<br />
j<br />
j<br />
nX<br />
1<br />
j D1<br />
D .n C 1/H n n: (14.27)
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