series and sequences math and science initiative

#Series and sequences math and science initiative series#

If we rotate just a bit further (to the green line L') and project the points onto this line, the order of the points 1 and 3 will reverse and we obtain the permutation 31452. As we rotate the line L into other positions in a counterclockwise direction, we reach a location where there is a pair of points in the original configuration which for the first time determine a line whose direction is perpendicular to the rotating line. If we take the point configuration above and project (using the red lines) its labeled vertices onto the blue line L, we obtain points in the order 13452.

$series and sequences math and science initiative$

To get at a concept of equivalence or inequivalence of point configurations that goes beyond the construction of associated graphs, let us consider the ingenious approach below, developed by Jacob Eli Goodman (City College, City University of New York) and Richard Pollack (Courant Institute of Mathematical Sciences, NYU). (They would still be isomorphic if we had included edges between the vertices 3 and 5.) However, can you make a case for why the two original point configurations might be said to be different, even though the constructed graphs were the same? Notice that as graphs, these two graphs are isomorphic. Since 5 points determine 10 lines, we might have 10 edges in the resulting graphs but we have chosen not to have an edge in diagrams for the line segments from the vertices labeled 3 and 5, because these segments would 'lie on top of' segments joining other vertices. The diagrams above were obtained from the original two-point configurations by joining any two vertices that determine a line with a line segment. The set of 5 planar points below will serve as an example. What is involved here is a collection of points (the points are labeled) at various locations in a plane. In the discussion below we see the complex way that previously explored concepts are put to work for the purpose of generating new ideas and making progress on problems that have eluded solution with previously explored ideas.Ī tenacious problem in dealing with configurations of points and lines has been finding a good way of determining when two such configurations are the same or different. It is subtle distinctions of this kind that intrigue mathematicians and drive math skeptics nuts! There are statements which are true of simple graphs but are not true of graphs with multiple edges. Digraphs can be thought of as an extension of the graph concept. There are pros and cons to either approach. Two schools have developed with regard to defining graphs so as to allow multiple edges and loops or not. Thus, allowing multiple edges and/or loops in a 'graph' extends the idea of a (simple) graph. Nearly always, something new is inspired by something old and extends some aspect of the old ideas in a new direction.

#Series and sequences math and science initiative series#

11.7: Absolute Convergence Roughly speaking there are two ways for a series to converge: (1) the individual terms get small very quickly, so that the sum of all of them stays finite, or (2) the terms do not get small fast enough, but a mixture of positive and negative terms provides enough cancellation to keep the sum finite.Often in the discussion of new ideas, new inventions, and new technologies, the distinction between incremental improvements versus true 'revolutions' is made.11.6: Comparison Test As we begin to compile a list of convergent and divergent series, new ones can sometimes be analyzed by comparing them to ones that we already understand.11.5: Alternating Series Next we consider series with both positive and negative terms, but in a regular pattern: they alternate.This means that if we can show that the sequence of partial sums is bounded, the series must converge. If all of the terms anan in a series are non-negative, then clearly the sequence of partial sums snsn is non-decreasing.

In many cases it is possible at least to determine whether or not the series converges, and so we will spend most of our time on this problem. 11.4: The Integral Test It is generally quite difficult, often impossible, to determine the value of a series exactly.A series converges if the sequence of partial sums converges, and otherwise the series diverges. Associated with a series is a second sequence, called the sequence of partial sums. 11.3: Series Recall that a series, roughly speaking, is the sum of a sequence.The range of the function is still allowed to be the real numbers in symbols, we say that a sequence is a function f:N→R.