Concrete mathematics : a foundation for computer science

264 SPECIAL NUMBERS
Rearranging, we have an expression for the difference between H, and Inn:
H,-Inn = 1- i(HF’ -1) _ f (j-$/%1) - $-$‘-1) - . . .
When n -+ 00, the right-hand side approaches the limiting value
1 -;&(2)-l) -3&(3).-l) - $(LV-1) -... >
which is now known as Euler’s constant and conventionally denoted by the
Greek letter y. In fact, L(r) - 1 is approximately l/2’, so this infinite series “Huius igitur quanconverges
rather rapidly and we can compute the decimal value titatis constantis
y = 0.5772156649. . . . (6.64)
Euler’s argument establishes the limiting relation
lim (H, -Inn) = y;
n-CC
(6.65)
thus H, lies about 58% of the way between the two extremes in (6.60). We
are gradually homing in on its value.
Further refinements are possible, as we will see in Chapter 9. We will
prove, for example, that
1 En
H, = lnn+y+&-- -
1 2n2 + 120n4 ’
O