Lec 39 | MIT 18.01 Single Variable Calculus, Fall 2007

The professor begins by acknowledging the support for MIT OpenCourseWare, which enables the free distribution of educational resources. He substitutes for Professor Jerison, who is in London, Ontario, and outlines the agenda for the day, which includes a review of power series and Taylor's formula, examples and applications, and a course evaluation survey. The professor also mentions a handout for the end-of-term and advises students to check the website for office hours and final exam details. He emphasizes the importance of understanding power series as generalized polynomials and introduces the concept of the radius of convergence, R, as a key factor in determining the series' behavior.

The professor explains the concept of the radius of convergence, stating that within this radius, a function behaves like a polynomial, having all its derivatives. He discusses Taylor's formula, which expresses the coefficients of a power series in terms of the function's derivatives at a point, typically zero. The professor then provides an example of the exponential function e^x, demonstrating how its power series is derived from its derivatives, all of which are e^x, evaluated at x = 0. He also addresses a question about the extent to which a power series must be written out to be considered complete, emphasizing the importance of recognizing the pattern.

The professor delves into the power series expansions for several common functions, such as sines, cosines, and tangents, which are all available in calculus. He provides a detailed example of the geometric series and its relation to the function 1/(1+x), highlighting the importance of the radius of convergence, which in this case is 1. The professor also discusses the implications of the graph of the function at x = -1, where the series does not converge, indicating the limit of the radius of convergence.

The professor continues with the topic of derivatives, showing how the power series for sine and cosine can be derived using Taylor's formula. He explains the periodic nature of the derivatives of sine and cosine and how they result in a repeating pattern of coefficients in their power series expansions. The professor emphasizes that the radius of convergence for the sine function's power series is infinity, meaning it converges for all values of x. He also explains the mathematical reasoning behind the convergence of the series terms to zero as n approaches infinity.

The professor discusses the properties of power series and the operations that can be performed on them, such as multiplication and differentiation. He provides an example of multiplying the power series of sine by x to obtain a new series, whose radius of convergence is the minimum of the two original series. The professor also explains how differentiation can be used to find the power series for cos(x) from that of sin(x), maintaining the same radius of convergence.

The professor explores the application of power series to integration, specifically finding the power series for ln(1+x) by integrating the known power series of 1/(1+t). He demonstrates the process of integrating term by term and evaluating the result at x and 0 to arrive at the power series expansion for ln(1+x). The professor notes that the radius of convergence for this series is 1, reflecting the behavior of the original function outside this range.

The professor concludes the lesson by discussing substitution in power series as a method to find new expansions. He uses the example of e^(-t^2) and shows how substituting x = -t^2 into the power series expansion for e^x yields a new series. The professor also introduces the error function, which is derived from the integral of e^(-t^2) and normalized by a factor of 2 over the square root of pi. He outlines the process of finding the power series expansion for the error function, emphasizing its importance in probability theory.

In the final paragraph, the professor wraps up the lesson and encourages students to take the next course, 18.02, which covers vector calculus. He suggests that this course will provide context for the material covered in the current course and explain certain concepts in more depth. The professor also mentions a poem by Professor Jerison that illustrates the unification of various calculus rules under the chain rule, emphasizing the interconnectedness of mathematical concepts.