"This was our last session, as the student's final for this class is on 12/19. We did an amazing amount of material in this class which would take a long time to summarize using a normal amount of accuracy. Therefore, here is an abbreviated list of things that we went over:
Approximations of functions using polynomials of a finite degree: Take as many derivatives of your function as you want in your polynomial, and plug in the value of the point you want to estimate into each of these derivative functions. That real number answer is the coefficient for a term in the polynomial. The overall polynomial will be of the form a sub zero plus a sub one times x plus a sub two times x squared over two factorial plus a sub three times x cubed over three factorial and so on.
Taylor Polynomials and Maclaurin Polynomials: These are Polynomials of a finite degree of the form described above, except they are centered about a point c, so in place of each x listed above should instead be x minus c. In an Maclaurin polynomial, c is always 0.
Power Series: A function can be represented EXACTLY as a power series, because it uses an infinite amount of terms to 'approximate' the function. The rule for convergence of a power series is that only one of the following can be true: Either the series converges only at the center, it converges for all x, or is converges for a distance from c equal to what's called the radius of convergence. The interval of convergence is (c-R, c+R). The ratio test is used to calculate the radius of convergence, and then the answer to the ratio test is compared with the form "the absolute value of x minus c is less than R in order to determine what R is equal to." We did many examples of how to do this.
Integration by parts: We reviewed the idea of IBP and how to do it. We then chose the hardest example in which two integrals have to be combined. The student did the hard example with only a couple of snags, and then did an easier version of the problem with ease! He showed great progress.
I also went through a review packet and he did some integrals from the packet before time was up. It was a fine ending to our time together."