What do you think is the most important concept to grasp about sequences and series?

Question

Also, which concepts are most mis-understood by students and then later gives them trouble later in more advanced classes. I am in Calc 2 now, but I plan on being a math grad student once I get my prerequisites. thanks for any relevant insight.

Answer 1

You always have to start with an easy tractable expression for T_n or S_n. If it ain't tractable, make it so. Avoid heavy algebra-crunching wherever possible, to let you concentrate on the form of the series (arithmetic, geometric, factorial, exponential, alternating, sum of separate components, other etc.)
- if you're not given closed-form for either of T_n or S_n, then calculate whichever one you need. (might involve using the standard sum-of-n^i formulae, or setting up a recurrence relation and polynomial solving)
- if you're given S_n and need T_n (or vice versa) then solve S_n - S_(n-1) = T_n, or S_n = Σ_i=0/1_i=n T_i
- if you're given an intractable form for T_n (or S_n), then find an easier form that you prefer to deal with (using the Comparison Test) e.g. T_n = (n)(n-1)(n+7)/4(n^4) is basically 1/4n in disguise. If you get factorials, exponents, other functions etc., then for convergence testing you can ignore the first few terms and compare the for whatever convenient n>N you choose. This can be useful to see asymptotic behavior.
- if you're given a reducible form for T_n (or S_n), then reduce it already! Typically things like T_n = cos(π/2 x) /x^n, noting cos(π/2 x) boils down to 1,0,-1,0,1,0,-1,...
becomes simply T_n = 1/x, 0, -1/x³, 0, 1/x^5, 0, -1/x^7, 0...

Answer 2

A good question that has attracted a wide range of considered answers on both sides of the debate apart from the usual crackpot ones. Causation is not so simple and I think, with all due respect to Aristotle, he does not have the final word on philosophy, just one world view and not that many followers at present. I notice a repeated logical flaw in some arguments. The CO2 coming before/behind a temperature change is a good example. Some here argue that CO2 is a greenhouse gas therefore increasing it in the atmosphere will logically increase temperatures. The amount of CO2 that ends up in the atmosphere is not closely linked to excess CO2 emissions by man as there are a huge variety of sinks etc that link into that. [oceans, plants, rest of carbon cycle]. Likewise it would be unrealistic to expect a close match between annual temperatures [even over decades] to match CO2 concentrations due to major fluctuations in ocean currents, seasons etc. But none of that would alter the underlying reasoning, increase GHGs you will eventually over the long term increase global temperatures. Others point out that in past global warming periods CO2 concentrations came after the warming and thus argue that CO2 can not be a contributor to GW. This clearly has 2 major flaws in it. 1] some previous GW episodes may well have had other understood causes, such as natural orbital and solo variations. In a warming period it is also known that there are positive feed back mechanisms, and melting perma frost will release CO2 as an example, so you would expect any warming to increase the concentration of CO2. This does in no way mean that CO2 does not or can not increase GW. In fact that is one of the issues facing us this time. This time we have pretty clear evidence that the CO2 concentration is increasing over time before warming, and it is a reasonable proposition that man's use of fossil fuels is the principle reason for that. Logic suggests that this will cause some warming. Past warming events suggest that with warming further CO2 is released. Further CO2 in the atmosphere will then create even further warming. Positive feedback. And even though it has been denied, a warmer planet will have more water vapour in the air and thus also have a positive feedback. There is no logic in saying that CO2 can not be both a cause and a result in this situation. There are natural limiting factors to the warming of course and we will not end up in a total runaway situation. There is a limit to the amount of carbon available to put into the atmosphere. It is not an endless supply. Water vapour will always remain in some equilibrium with its liquid state. As the earth heats up it will radiate more energy and all that the extra GHGs will do is let it reach a new higher equilibrium state. For humans of course the concern is just what this final state will be, because the impacts of even a few degrees warmer is huge.

Answer 3

Probably *the* most important concepts about series and sequences are convergence and divergence (and how to test for them). But it's also important to be able so recognize sub-sequences and sub-series as well (e.g. seeing that e^iΦ = cos(Φ)+i*sin(Φ) and so on).

HTH

Doug