A limit

Félix Locas presented me this problem.

Let r(n) = \lfloor \log_2 \frac{n}{\log_2 n} \rfloor. Show that

\lim_{n \rightarrow \infty} \left( \log 2+\sum_{k=1}^{r(n)} \frac{1}{k(k+1) 2^k} \right)^n = 1.

My solution

The series \sum_{k=1}^{\infty} \frac{1}{k(k+1) 2^k} is easy to calculate. It is, for instance, the difference between the integrals of geometric series:

\sum_{k=1}^\infty \frac{1}{k(k+1) 2^k} = \sum_{k=1}^\infty \frac{1}{k 2^k} - \sum_{k=1}^\infty \frac{1}{(k+1) 2^k} = 1-\log 2.

Furthermore, abbreviating r = r(n),

r^{3/2} 2^{r} \sum_{k=r+1}^\infty \frac{1}{k(k+1) 2^{k}} \le \sum_{k=0}^\infty \frac{1}{\sqrt{r} 2^k} \xrightarrow{r \rightarrow \infty} 0

implies that for n sufficiently large we have \sum_{k=r+1}^\infty \frac{1}{k(k+1) 2^{k}} < r^{-3/2} 2^{-r} and

\log 2+\sum_{k=1}^{r(n)} \frac{1}{k(k+1) 2^k} = 1-\sum_{k=r+1}^{\infty} \frac{1}{k(k+1) 2^k} \geq 1 - r^{-3/2} 2^{-r}. \qquad (*)

Finally, since r = \log_2 \frac{n}{\log_2 n} - \varepsilon_n for some 0 \le\varepsilon_n < 1, we have

n r^{-3/2} 2^{-r} = \frac{2^{\varepsilon_n}\log_2 n}{(\log_2 n - \log_2 \log_2 n - \varepsilon_n)^{3/2}} \rightarrow 0

which implies that

\left( 1 - r^{-3/2} 2^{-r} \right)^n \rightarrow 1.

Since also \log 2+\sum_{k=1}^{r(n)} \frac{1}{k(k+1) 2^k} \le 1 comparing this with (*) yields

\lim_{n \rightarrow \infty} \left( \log 2+\sum_{k=1}^{r(n)} \frac{1}{k(k+1) 2^k} \right)^n = 1.

Poster critique

I printed my poster today (see this post) for the 11th Bayesian nonparametrics conference. Here’s the final version (42in x 60in).

poster_final.png

While I was first satisfied with it, as it appeared on my computer screen, staring at it full size changed my perception.

Before jumping into the critique, let me list a few things I like about the poster.

  1. The content, which has been thoroughly reviewed.
  2. The text size (44pt). It is clear and easy to read between 3 to 9 feet away.
  3. It is self-contained and explains itself.
  4. I left plenty out. I listed the two main contributions of our work and that’s what I talk about.

Now for the things I dislike.

  1. The titles are too small.
  2. Using letters in both calligraphic and normal font. This makes it harder to refer to the poster.
  3. It looks messy. The structure of the poster should be graphically emphasized rather than follow columns of text.
  4. The white background is unattractive; the title banner is bland.
  5. “References” should be replaced by “Works cited”.
  6. Should I talk more about Bayesian procedures / posterior simulation? I could have shown posterior mean estimates, the graphs of densities smoothed using our operators, etc.

In short, I have mixed feelings about the poster. On one hand, I like that the text is large, that it is self-contained and that it goes pretty much straight to the point. There are no meaningless graphics under a “result” banner. On the other hand, it looks messy and I’m afraid people will get lost in it.

Next one will be better.