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 ninfinity Becky 

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. 

