Quandaries and Queries


I am a high school teacher (grades 9 - 12) and have a question about harmonic numbers.

What can you tell me about the limit of harmonic numbers as it reaches infinity?

lim Hn n-infinity




Hi Becky,

I presume you mean Hn = 1 + 1/2 + ... + 1/n. As n increases, Hn increases and is unbounded. That is, if you give me a number M, no matter how large, I can find an n so that Hn > M. Thus Hn -> infinity as n goes to infinity.

One way to prove this by comparing it to the integral of (1/x) between the limits of 1 and n. This tells us the answer is that Hn grows like log(n) and thus Hn -> infinity as n goes to infinity.



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