Question: Without using calculus (as in the previous exercise), show that, if n is a power of 2 greater than 1, then, for H n ,

Without using calculus (as in the previous exercise), show that, if n is a power of 2 greater than 1, then, for Hn, the nth harmonic number, 

Hn ≤ 1 + Hn/2

Use this fact to conclude that Hn ≤ 1 + [log n], for any n ≥ 1.


Data From Previous Exercise

Use the fact that, for a decreasing integrable function, f,

cb+1 b f(x)dx < f (x)dx, r=a r=a-1 2=a 

to show that, for the nth harmonic number, Hn, 

ln n ≤ Hn ≤ 1 + ln n.

cb+1 b f(x)dx < f (x)dx, r=a r=a-1 2=a

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