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Transcript

Calculus Notes Grinshpan 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.