Available online
Description
This programme looks at the generation of new, convergent, series from old, by alternating the signs of terms, and introduces the idea of a power series.
This programme looks at the generation of new, convergent, series from old, by alternating the signs of terms, and introduces the idea of a power series.
| Module code and title: | M203, Introduction to pure mathematics |
|---|---|
| Item code: | M203; 03A |
| First transmission date: | 03-05-1987 |
| Published: | 1987 |
| Rights Statement: | |
| Restrictions on use: | |
| Duration: | 00:24:45 |
| + Show more... | |
| Producer: | Pip Surgey |
| Contributors: | Ian Harrison; Keith Hodgkinson; Roy Nelson |
| Publisher: | BBC Open University |
| Keyword(s): | Clarinet; Convergent; Divergent; Harmonic; Series; Synthesizer; Tuning fork |
| Footage description: | When is it possible to add up an infinite number of real numbers and get a finite answer? Does it make sense to add up an infinite number of functions? What do you get if you do? This programme introduces the idea of a power series by considering these questions, and then goes on to define functions like e, cosh x, sinh x, sin x and cos x by their power series. It begins by comparing a pure note, as given by a tuning fork, with the same note played on a clarinet, and demonstrates, using a synthesizer, that whereas the pure note can be represented by a sine wave, the note from the clarinet includes overtones, so is the sum of sine waves of different frequencies and amplitudes. Next the programme discusses the series and (the harmonic series) with the help of animations, finding that whilst the first two are convergent, the last is divergent. However, by alternating the signs of the terms, all three new series are convergent by the Alternating Test. (...) By considering the sequence of partial sum functions by an animation, it looks as if it is the graph of an exponential function. With the aid of a micro, it is shown that this power series, in fact, sums to ex for all values of x. Finally, the programme generates the power series for cosh x and sinh x, and, by alternating the signs of the terms of these, those for cos x and sin x. The programme uses many animations to illustrate sequences of partial sums and hence convergence, or otherwise, of both numerical and power series, and so it should be of interest to students and teachers alike who are interested in mathematics at either advanced or undergraduate level. |
| Master spool number: | HOU5627 |
| Production number: | FOUM269W |
| Videofinder number: | 2377 |
| Available to public: | no |
