175 -A curious series

November 10, 2008

I just learned from the textbook that apparently whether the series

\displaystyle \sum_{n=1}^\infty\frac1{n^3\sin^2 n}

converges is still open, which I find rather surprising. The reference the book lists is the book Mazes for the Mind by Clifford Pickover, St. Martin Press, NY, from 1992, but Dr. Pickover has informed me that he believes the problem is still unresolved; he also discusses it in his book The Mathematics of Oz, Cambridge University Press, 2002. I would be very curious to hear from updates or suggestions, if you have any.

Here is a slightly technical (and very quick, and not particularly deep) observation: The issue seems to be to quantify how small \sin^2 n is, when it is small, or more precisely, how sparse the set of values of n is for which the sine function is “significantly small.” One could start by looking at n so that |n-m\pi| is small for some m, so we are led to consider the standard convergent approximations to \pi, satisfying \displaystyle \left|\frac nm-\pi\right|<\frac1{m^2}. This means that 1/(n^3\sin^2 n) is close to, but slightly larger than, \displaystyle\frac{1/\pi^2}n, and so the question leads us to the problem of how sparse the sequence of numerators of the rational approximations to \pi actually is, something about which I don’t know of any results.

Below I display some graphs for the partial sums of the series. Let \displaystyle S(n)=\sum_{k=1}^n\frac1{k^3\sin^2 k}. The first graph shows n vs. S(n) for 1\le n\le 100. In the other graphs, n goes up to 300, 1000, and 100000. (Thanks to Richard Ketchersid for the code.) It is not clear to me that the last graph is accurate or that it allows us to draw any conclusions (it certainly seems to suggest that the series converges to a number slightly larger than 30); it may well be that further jumps are beyond the range I chose, or that the approximations Maple uses in its computations are not fine enough to examine very large values of the series.

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175, 275 -Homework 10 and suggestions for next week

November 10, 2008

Homework 10 is due Tuesday, November 18, at the beginning of lecture. The usual considerations apply.

In 175 we will try to cover this week until section 8.7 at least; it is possible this won’t happen until next week. Sections 8.3, 8.4, and 8.5 all introduce important tests for convergence of series; make sure you understand the arguments being presented (rather than just trying to memorize the tests). The material in section 8.6 is particularly important (conditional and absolute convergence). We will also cover some additional material on the p-series \displaystyle \sum_{n=1}^\infty\frac1{n^p}

Next week we will continue from section 8.7 (or at whatever point in the chapter we find ourselves at that point) on. We will also cover a few additional topics that the book doesn’t mention or doesn’t treat in sufficient detail, once we are done with 8.10. If you want to read ahead, the topics we will cover are uniform vs. pointwise convergence (including Wierstrass test) and infinite products. I will distribute notes of the topics not covered in the book.

In 275 we will cover from section 13.6 on, but the emphasis will be on section 13.8, and the notion of Jacobian. We will also present a few notions from linear algebra to make sense of the general version of the chain rule. Afterwards, we will continue with Chapter 14, which contains the main results from this course you are likely to use in the future. Chapter 14 will take some time to cover. 

Homework 10:

175: Do not use the solutions manual for any of these problems. 

  • Turn in the problems listed for Homework 9  that you still have pending.
  • Section 8.4. Exercises 3, 12, 23, 26, 35, 38, 40.
  • Section 8.5. Exercises 4, 8, 10, 25, 31, 34, 44, 47.

Besides the exercises you have pending from last week, there are 15 new problems. Turn in the exercises you have pending, and at least 7 of the new problems. The others (at most 8) will be due December 2, together with the additional exercises for that week. 
275:  

  • Section 13.5. Exercises 4, 8, 19, 22, 30, 45.
  • Section 13.6. Exercises 6, 11, 24, 30.
  • Section 13.7. Exercises 13, 21, 37, 60, 78, 79.
  • Section 13.8. Exercise 1, 6, 16, 21.

There are 20 exercises this week. Turn in at least 10. The remaining problems (at most 10) will be due together with a few additional exercises on December 2.