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
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.
Penny