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FEMALE VOICE: The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation, or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Hi. Today, we're going to learn a rather powerful technique, called the technique of partial fractions, that's particularly applicable for one special type of integrand. In particular, it's going to be applicable to the situation where the integral has the form, the quotient, of two polynomials. In other words, suppose we have integral ''P of x' dx' over 'Q of x', where 'P' and 'Q' are both polynomials in 'x'. And, by the way, for reasons that will become apparent later, we'll assume that the degree of 'P' is less than the degree of 'Q'. In other words, that the highest exponent...

FEMALE VOICE: The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation, or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Hi. Our lecture today concerns mathematical induction, which, roughly speaking, is a technique that one uses to prove something when one already has a pretty good suspicion as to what the right answer is. Now, rather than to philosophize about this too long, let's tear right into a problem and see, in action, just what the concept means. Recall that, in an earlier lecture, we have proven that the limit of a sum is equal to the sum of the limits provided, of course, that there are only two terms in the sum. That is, if we have two functions, 'f 1' and 'f 2', the limit of 'f1 of x' plus 'f2 of x' as 'x' approaches ...