Harmony in Infinite Series

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 S_n 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 S_2ⁿ 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 H_n 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, H_n? 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?