(A sequence is a bunch of items, separated by comma: For example: 1,3,5,7,9,...
A series is the sum of such items: 1+3+5+7+9...)
The theme of calculus is using a breakthrough concept of the infinitely small. We use infinitesmal small changes in y over x to calculate the slope of the tangent line, and integrating area under curve by adding infinitely small rectangles. But it doesn't make sense to be adding up infinitely many things, even if they are small. In order for this to be fully defined, we need the limit.
The topic of sequence and series are often seen as an application of calculus and it also relies on limits.
But, why study sequence and series? Who cares if a certain series converge or diverge?
I see no need for regular calculus 101 and 102 students worry about this.
For the math aficionado, one may find the condition for divergence interesting. For the ever increasing odd series above, of course it diverges. But not all ever decreasing series converge, consider the following 2 series:
1+1/2+1/3+1/4+1/5+...
1+1/2+1/4+1/8+1/16+...
Both are decreasing, one may guess both will eventually converge. But no, the first one (known as the harmonic series) diverge. We can write a simple computer program to add a few hundred terms to see that. In the second series. each term is multipled by a factor smaller than 1, and one can show such cases always converge (these are known as geometric series).
What is really interesting is the following series:
1+1/22+1/32+1/42+1/52+...
This series converges... to π2/6! How does this relate to π? Oh my. Write a little program to verify this yourself. Ok, I give you the little routine.
public void zeta2() {
double sum = 0;
for (int i=1; i < 300; i++) {
sum += 1.0/(i*i);
}
System.out.println("total="+sum);
}
This series is known as the Zeta function. It is lengthly and no so trivial to show how this relationship work, but thanks to the internet, it is readily available. This is usually not covered in a typical calculus book, however THICK.
The neat stuff of math is what students need to see, not mundane boring exercises. Non-math majors should also be shielded from the ultra technical math formalism too. I suggest either take out sequence and series altogether from calculus curriculum or add this topic. This topic is my most recent entry in my book.
3 comments:
Math topics are fascinating, but each and every of the thick math books mentioned need to be broken down into much smaller volumes to be less intimidating. There need to be ample back stories about the math guys of the past and real applications of these head knowledge. Math is an art, it's got history, it's got elegance, should teach math that way.
The stupid application for the integral is to ask in every imaginable way of the same question, and the students makes an equation to equal 0. (to find the max and min of a curve)
Math books has sadly become a list of theorems and the 'A' student would spot the patterns and apply the meaningless theorems and solve the problems. The B/C students are labeled losers and they are not.
D students? maybe.
um, derivative = 0 to get max or min. derivate = 0 means no change at that point, so it is max or min.
calculus at least has some application.
Hardcore math have very little or no applications(!). Ok, at least not very easy to see how they can be applied.
Volumes have been written on history of math guys. Each guy has their own amazing legacy
you're right, I mean derivative = 0. All the word problems point to this single use for calculus, find the max and min given a function that describe the problem.
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