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I have the following harmonic sequence:

h(n) = 1 + 1/2 + 1/3 + 1/4 +...+ 1/n

Id like to prove that there's a recurrence with

h(n) (less than or equal to) h( lowerbound( n/2)) + 1
Bacteria
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Justin T.
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  • Are you assuming that h(n) <= h(floor(n/2)) + 1 and trying to use that to construct a recurrence, or trying to prove h(n) <= h(floor(n/2)) + 1 ? – Jerry Federspiel Sep 01 '15 at 20:00

1 Answers1

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This belongs on math.SE, but we have

h(2n) - h(n) = 1/(n/2 + 1) + 1/(n/2 + 2) + ... + 1/n
             < 1/(n/2) + 1/(n/2) + ... + 1/(n/2)
             = 1,

since there are n/2 terms. I'll leave the odd case as an exercise.

David Eisenstat
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