2007-10-15 - Properties of sequences

Today's nonsense
- Let's do an unofficial count. Dakota has not seen The Princess Bride yet. Alright, we're done. Soon we're going to have to kick him out of the room, and with Katie, until they see the movie.
- Dakota: "You can't really cheat on a math test."
- "How do you do a capital 'b' in cursive?" - Dakota, week three on cursive note-writing.
- AntiAvery Day today
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Definition of a monotonic sequence- A sequence {a_{_n}} is monotonic if its terms are nondecreasing, so that a_1 <= a_2 <= a_3 ... <= a_n <= .... or if terms are nonincreasing (a+1 => a_2 => a_3 => ... => a_n => ...) Monotonic means you have no relative maximums and no relative minimums. So you can take the first derivative, and if there's no sign change, then that means that the sequence is monotonic. A parabola cannot be monotonic.

ex) Determine whether the sequences are monotonic.
a) a_n = 3 + (-1)^n ---> Not monotonic. You can't prove this just by showing the first four terms ... you have to show it based off of the equation of the sequence itself.
b) a_n = (2n)/(1+n) ---> There are two ways to prove that this is or is not motonic- either algebraic or by calculus. By algebraic, you compare 2n/(1+n) to 2(n+1)/(1+(n+1)) ... and then you show which one is greater and which one is less, so evaluate that out and cross multiply etc.
-- The other method: take the derivative. Yes, the very derivatives that we have spoken of in the past. My derivative comes out to be ... Lodehi - Hidelo all over lolo. So what's the numerator? What does the numerator come out to be? 2. Yes. Ahuh. Ahuh, oh yes you did. I understand. It's okay. You guys started off not talking, did you hear the griping about what we were doing? And then some people are paying attention. Do you see the derivative? "Who started the Princess Bride thing?" - Dakota. Anyway, if the derivative is always going to be increasing or decreasing, then you know it's monotonic. So, yes, it's monotonic, and we proved it two different ways.

Definition of a bounded sequence- A sequence {a_{_n}} is bounded above if there is a real number M such that a_n <= M for all n ... the number M is called an upper bound of the sequence.
--- A sequence {a_n} is bounded below if there is a real number N such that N <= a_n for all n. The number N is called a lower bound of the sequence.
If there is a largest number in your sequence, then it's upper bound, and if there's a lowest number in the sequence, then there's a limit-bound that we don't cross. If it has an upper and lower bound, then it's a bounded sequence. You can find one of the bounds by taking the limit, in many cases.

Also, 3. A sequence {a_{_n}} is bounded if it is bounded above and bounded below.

Bounded monotonic sequences-- If a sequence {a_{_n}} is bounded and monotonic, then it converges. If you have an upper bound AND lower bound, and it's monotonic, then it converges to whatever that upper or lower bound is. But you can't say that "if it converges, then it's monotonic," which is not true. In the next chapter we will talk about series. Sequences is the numbering pattern. Series is talking about the sum of all of the terms. The sum of the sequence is the series.

ex) Determine whether each sequence is monotonic. Discuss the boundedness of the sequence.
1) a_{_n} = 4 - (1/n) .... monotonic, so it's bounded (and therefore upper and lower) (just for yourself: both 3 and 4 are the bounds)
2) a_{_n} = (cos n)/n ..... not monotonic because cosine goes up and down .. so it's bounded at 1/n and another one, so this is going to be a cosine graph that gets smaller and smaller. You'd have to plug in the first couple terms to figure out the actual bounds, of course.
3) a_{_n} = (2n^{^2} - 1) / (n+3) .... monotonic and lower bounded. How do you know that it's monotonic? Bounded says whether or not there's a limit to how big or small your numbers get. The monotocity? Use the quotient rule.

"Sensible Calculus" / Sequences (M. Flashman, 2005)
Math notes on sequences (Paul Dawkins, 2003-2007)
Short introduction to sequences (Mohamed A. Khamsi, 1996)
EquationSheet re: sequences & series
Wikipedia page on sequences
Relevant Wikibooks page on sequences