Day 28 in Mat229

• The quiz and Key
• Median: 8.5; Mean: 8
• Comments:
  1. Because you were integrating a quadratic, Simpson's Rule will be exact.

     The way that Simpson is designed, it means that the error in the midpoint method should be exactly half the error of the trapezoidal method. Is yours? If not, that's a sign that you've made an error....

  2. If the function is increasing, then lrr is an under, and rrr an over; vice versa if decreasing.

     Trap and Mid depend on concavity (since both methods can be thought of as areas of trapezoid -- connecting points for trapezoidal, passing through midpoints as tangent lines for midpoint.

     Concave up implies trap is over, mid is under.

  3. In the end, \(n\) must be an integer. So use "Ceiling" to round up to the nearest integer.

     If you plan to use Simpson's rule when you're done, as an average of trap and mid, then you would need to use the same number of rectangles for both. In that case, you'd choose to use 37 rectangles (rather than midpoint's lower value of 26).

The chaos, conflicting information, firings, and hurtful rhetoric of the Trump administration's approach to science over the past month are causing anxiety, grief, and concern for the scientific community in the United States. The dramatic events are reverberating around the globe, especially among international scientific collaborators. Not surprisingly, scientists in the US and around the world do not agree on the best approach to preserve science during this onslaught. A diversity of thought has always been a strength of the scientific enterprise but at the same time, there are principles around which all scientists unite-those of evidence, independence, process, and inclusion. These common values must now propel everyone in the scientific community to work together as never before to stand up for science....

Those in the scientific community who enjoy the protections of academic and other freedoms afforded by the US Constitution's First Amendment should do and say more. (My emphasis) Some will march in the streets, some will send messages to Congress, and some will focus on their research, students, and trainees. And just as leaders deserve some leeway to decide when to speak out most effectively, they should support their constituencies by not trying to squelch or steer more strident voices.

(By the way, do you know that there are different sized infinities? "Countably infinite" is the smallest. What's the largest?)

One famous sequence is the Fibonacci sequence, which is defined recursively by naming the first two terms in the sequence explicitly and then describing a pattern for the rest of the terms: \[ \begin{array}{l} {F_1=1}\cr {F_2=1}\cr {F_n = F_{n-1}+F_{n-2} \ \ n \ge 3} \end{array} \] This sequence appears throughout nature, and is named for the 13^th century Italian mathematician Fibonacci Pisano.

• The sequence stems from a rabbit problem:
• Let's see where they came from, and why they're so special.
• They are the "worst case scenario" for the Euclidean algorithm to find the GCD (greatest common divisor) of two numbers, which we will investigate with this handout.
• The ratio of successive Fibonacci numbers approaches the golden mean as \(n \to \infty\): \[ \varphi = \frac{1+\sqrt{5}}{2} \approx 1.618 \]

In other words, a sequence is convergent to \(L\) if we can make the difference between the terms and the limit \(L\) as small as we like (\(\epsilon\)) by merely looking far enough down the road (from \(N\) on).

Hence when I say that "The ratio of successive Fibonacci numbers approaches the golden mean.", what I mean is that the sequence of ratios of successive Fibonacci numbers, \(a_n \equiv \frac{F_{n+1}}{F_n}\), converges to golden mean: \[ \lim_{n \to \infty} a_n=\varphi \]

Hence we can say that the sequence of Fibonacci numbers themselves diverges to infinity: their ratios converge, but the numbers are growing -- exponentially, actually, and a closed-form formula for them is given by \[ F_n = Round\left[\frac{\varphi^n}{\sqrt{5}}\right] \]

What I'm calling a "closed-form solution" (or formula) our authors would call the "an explicit formula for the \(n^{th}\) term of the sequence."

I really don't like problems of the form of Example 5.1 and Checkpoint 5.1! (The correct answer to those is "What do you want it to be?")

The 5.2 example and checkpoint are fine, because we've made clear what the pattern is. Using ellipses (...) is dangerous -- we don't know what happens next, for sure....

A sequence $\{a_n\}$ is called bounded above if there is a number $M$ such that \[ a_n \le M \ \ \ \ \ {\textrm{for all}} \ n \ge 1 \] It is called bounded below if there is a number $m$ such that \[ m \le a_n \ \ \ \ \ {\textrm{for all}} \ n \ge 1 \] If it is bounded both above and below, then it is a bounded sequence.

Hence the sequence of ratios of Fibonacci numbers, \(\{\frac{F_{n+1}}{F_n}\}\), is bounded above (by 2), and bounded below (by 1). The sequence \(\{\frac{F_{n+1}}{F_n}\}\) is a bounded sequence.

It is not monotonic, however. The values bounce above and below the limit, which is \(\varphi\).

• Theorem 5.1: If $\displaystyle \lim_{x \to \infty} f(x) = L$ and $f(n)=a_n$ when $n$ is a natural number, then $\displaystyle \lim_{n \to \infty} a_n = L$.

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  It turns out that the \(n^{th}\) Fibonacci number \(F_n\) is given exactly by \[ F_n=\frac{\varphi^n - (-\varphi)^{-n}}{\sqrt{5}} \] That's an explicit function of \(n\).

  While this function isn't real-valued defined for some powers of \(x\), we can formally take a limit of \(\{\frac{F_{n+1}}{F_n}\}\), and we'll see that it approaches \(\varphi\): \[ \lim_{n \to \infty}\frac{F_{n+1}}{F_n} = \lim_{n \to \infty}\frac {\frac{\varphi^{n+1} - (-\varphi)^{-(n+1}}{\sqrt{5}}} {\frac{\varphi^n - (-\varphi)^{-n}}{\sqrt{5}}} \] We can eliminate the \({\sqrt{5}}\), and pull a factor of \(\varphi\) from the numerator: \[ \lim_{n \to \infty}\frac {\varphi^{n+1} - (-\varphi)^{-(n+1)}} {\varphi^n - (-\varphi)^{-n}} = \lim_{n \to \infty}\varphi \frac {\varphi^{n} - (-1)^{n+1}\varphi^{-(n+2)}} {\varphi^n - (-1)^n\varphi^{-n}} \] which, upon dividing top and bottom by \(\varphi^n\), becomes \[ \lim_{n \to \infty}\varphi \frac {1 - (-\varphi)^{-(2n+2)}} {1 - (-\varphi)^{-2n}} = \varphi \lim_{n \to \infty} \frac {1 - \frac{(-1)^{n+1}}{\varphi^{2(n+1)}}} {1 - \frac{(-1)^n}{(\varphi)^{2n}}} = \varphi \] We actually rely on the following theorem not explicitly mentioned in the text to finish that argument, which I hope you will accept intuitively!

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• Theorem: If $\displaystyle \lim_{n \to \infty}|a_n| = 0$, then $\displaystyle \lim_{n \to \infty}a_n = 0$.
• Theorem 5.2: If $\{a_n\}$ and $\{b_n\}$ are convergent sequences and $c$ is a constant, then \[ \begin{array}{l} {\lim_{n \to \infty} c = c}\cr {\lim_{n \to \infty} ca_n = c\lim_{n \to \infty} a_n}\cr {\lim_{n \to \infty} (a_n + b_n) = \lim_{n \to \infty} a_n + \lim_{n \to \infty} b_n}\cr {\lim_{n \to \infty} (a_n - b_n) = \lim_{n \to \infty} a_n - \lim_{n \to \infty} b_n}\cr {\lim_{n \to \infty} (a_n b_n) = \lim_{n \to \infty} a_n \cdot \lim_{n \to \infty} b_n}\cr {\lim_{n \to \infty} \frac{a_n}{b_n} = \frac{\lim_{n \to \infty} a_n}{\lim_{n \to \infty} b_n} \ \ {\textrm{if}} \ \ \lim_{n \to \infty} b_n \ne 0} \end{array} \]

  Theorem 5.3 (Continuous Functions Defined on Convergent Sequences): Consider a sequence $\{a_n\}$ and suppose there exists a real number \(L\) such that the sequence $\{a_n\}$ converges to \(L\). Suppose \(f\) is a continuous function at \(L\). Then there exists an integer \(N\) such that \(f\) is defined at all values \(a_n\) for \(n \ge N\), and the sequence $\{f(a_n)\}$ converges to \(f(L)\).

  

• Theorem 5.4 (Squeeze Theorem): If $a_n \le b_n \le c_n$ for $n \ge n_0$ and $\displaystyle \lim_{n \to \infty}a_n = \lim_{n \to \infty}c_n = L$, then $\displaystyle \lim_{n \to \infty}b_n = L$.
  

  The "$n \ge n_0$" part says that the squeeze has to be on eventually, but not necessarily from the outset. It's the tail of the sequence that's important for the convergence, not the head.

• Theorem 5.5 (Convergent Sequences Are Bounded): If a sequence $\{a_n\}$ converges, then it is bounded.

• The sequence $\{r^n\}$ is convergent if $-1 < r \le 1$ and divergent for all other values of $r$: in particular, \[ \lim_{n \to \infty}r^n = \left \{ \begin{array}{lc} {0}&{-1 < r < 1}\cr {1}&{r=1} \end{array} \right . \]

• Theorem 5.6 (Monotone Convergence Theorem): Every (eventually) bounded, monotonic sequence is convergent.

  The point of adding "eventually" is that it is only the tail of a sequence that matters for convergence: what happens for any finite bunch of terms at the beginning won't have any bearing on the convergence as \(n \to \infty\).

Sequences are like the natural numbers (1, 2, 3, $\ldots$.): they have distinct ordered terms traipsing off into the far distance. We're interested in what happens as the terms traipse off. Do they approach a fixed value? Do they oscillate, bouncing back and forth? Do they get larger and larger, or smaller and smaller? Several interesting example are included, such as the Fibonacci numbers (which came about from a silly rabbit population story problem in Fibonacci's Liber Abaci).

In this section we encounter many definitions, and a few theorems which help us to understand when a sequence converges (its terms approach a fixed value), or diverges (doesn't converge!). This is an issue of fundamental importance as we push on to our major objective: representing a function using an infinite sequence of functions! We start with numbers, of course, because that's a simpler case.

So the analogy we're working from is that \(\varphi\) can be approximated arbitrarily well by a sequence of ratios of Fibonacci numbers \(\{\frac{F_{n+1}}{F_n}\}\) as \(n \to \infty\).

We'll soon be doing the same thing for functions, rather than \(\varphi\)....