To continue our discussion of infinite series, I usually show students the famous proof that the harmonic series 1+1/2+1/3+1/4+... diverges. This series is paradoxical to students because, in their minds,there is convergence, since the terms themselves approach zero. With some exploration they can begin to appreciate that convergence of the sum of the terms depends on how fast the terms approach zero! Most of the content of the student investigation below can be found in MathWorld or Wikipedia but my intent, as it almost always is on this blog, is to produce a classroom experience for students and an activity for teachers to use, not just an expository piece of writing.
Consider the following "S-series":
1 + 1/2 + 1/4 + 1/4 + 1/8 + 1/8 + 1/8 + 1/8 + 1/16 + 1/16 +...
(a) Continuing this pattern (of repeating groups of reciprocals of powers of 2), what would the 16th term be?
(b) If Sn represents the sum of the first n terms of this series (where n is a positive integer), what is the value of S16? No calculator!
(c) Develop a formula for S2n and verify your formula for S1024. Here, n = 0,1,2,...
(d) What conclusion can you draw about the convergence of the "S-series?" Explain.
(e) Consider the harmonic series (which we will call the "H-series"):
1 + 1/2 + 1/3 + 1/4 + 1/5 + ... + 1/n + ...
Let Hn represent the sum of the first n terms of this series.
Show that H16 > 3, H1024 > 6 and H65536 > 9 by comparing the "H-series" to the "S-series" term by term.
(f) Based on the above, what conclusion can you draw regarding the limit of the sequence of partial sums, Hn? What does this imply about the convergence or divergence of the harmonic series? How would you describe the rate at which this series converges or diverges?
(g) Research the harmonic series online. Be prepared to answer the following question:
What does the harmonic series have to do with overtones in music?
(h) Consider generalizations of the harmonic series, such as replacing 1/n by 1/(kn+j). Make two such generalizations and examine convergence in each case.
The possibilities are endless. If two roads diverged in the woods, which one would you take?
Friday, April 13, 2007
Harmony in Infinite Series
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5 comments:
A further generalization is to take the series 1/p(n), where p(n) is the nth prime. I am always amazed by the fact that this series also diverges.
TC
Wikipedia is featuring an interesting article on summability today.
thanks tc and eric--
The sum of the reciprocals of primes is certainly paradoxical, although the distribution of primes according to the Prime Number Theorem is related to the function n/ln(n) so, perhaps, it isn't so strange.
Eric,
That is a fascinating article. I will direct my students to read it. I already mentioned Cesaro sums to them and the various sums one could obtain depending on one's definition of summability and level of rigor.
Actually, the divergence of the sum of the reciprocals of the primes is quite natural, and is analogous to the harmonic series via the integral test.
A simple consequence of the prime number theorem,
&pi(n) ≈ n / ln n
is the estimate
p_{n} ≈ n ln n.
Thus, the convergence of ∑ 1/p_{n} is of the same character as the convergence of the integral ∫ dx/(x ln x). Since the indefinite integral is ln ln x, and that tends to infinity, though incredibly slowly, the series also diverges. I wonder what the analog of γ is.
eric--
Thanks for clarifying my previous comment on the Prime number Theorem! A very simple clear argument...
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