I'm getting ready to teach the second calculus course in the 4 course sequence at my school. One of the required topics is an introduction to number sequences. I want to motivate this section with some interesting examples from our physical universe. I plan on including the Fibonacci sequence:
\begin{align} F(n+2) = F(n+1)+F(n);\qquad F(0)=1, F(1)=1 \end{align}
This one shows up in many places, Greek architecture, various plant life, rabbits! etc. I was wondering if any of you had some other interesting sequences with physical significance that would be nice for a calc 2 class discussion. Thx
How about the expected number of record annual high temperatures recorded in a given city -- with relevance to the possibility of climate change. If the temperature observed in a year has no dependence on the temperature observed in previous years, then the expected number of record highs in n years is:
$$H_n = \sum_{k=1}^{n} \frac{1}{k},$$
the sequence of partial sums of the harmonic series. So now you bring up sequences and series.
The expected number of records in the first year is by definition $1$. Given independence, the probability that the second year is a record is $1/2$ -- so the expected number of records for $2$ years is $1 + 1/2$. The probability that the thrd year is a record must be $1/3$ (look at the possible orderings with equal likelihood), and the expected number of records is now $1+1/2+1/3$, etc.