Download THE PARTIAL SUMS OF THE HARMONIC SERIES

Calculus Notes
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THE PARTIAL SUMS OF THE HARMONIC SERIES
The series
∞
X
1 1
1
1
= 1 + + + ... + + ...
n
2 3
n
n=1
is called harmonic, it diverges to infinity. Its partial sums
1
1
Hn = 1 + + . . . + , n = 1, 2, 3, . . . ,
2
n
(harmonic numbers) form a monotone sequence increasing without bound.
The integral estimates
1+
1
1
+ ... + >
2
n
Z
n+1
1
dx
= ln(n + 1)
x
and
Z n
1
dx
1
+ ... + <
= ln n
2
n
x
1
are justified geometrically. Combined together, they give
ln(n + 1) < Hn < 1 + ln n,
n > 1.
Therefore Hn tend to infinity at the same rate as ln n, which is fairly slow. For
instance, the sum of the first million terms is
H1000000 < 6 ln 10 + 1 ≈ 14.8.
Consider now the differences δn = Hn − ln n. Since
ln(1 + n1 ) < Hn − ln n < 1,
n > 1,
we conclude that every δn is a positive number not exceeding 1.
Observe that
δn − δn+1 = (Hn − ln n) − (Hn+1 − ln(n + 1))
1
= ln(n + 1) − ln n −
n+1
Z n+1
1
dx
−
>0
=
x
n
+
1
n
(draw a picture to verify the last inequality). So δn > 0 are monotone decreasing.
By the Monotone Sequence Theorem, δn must converge as n → ∞. The limit
γ = lim δn = lim (Hn − ln n)
n→∞
n→∞
is called the Euler constant (Euler, 1735), its value is about γ ≈ .5772.
Thus, for large n, we have a convenient approximate equality
1
1
Hn = 1 + + · · · +
≈ ln n + γ.
2
n
It is not known to this day whether γ is rational or irrational.