"The student and I went over infinite series this week. We began with a review of what a sequence is and the two basic different types of sequences, arithmetic and geometric. We reviewed that an arithmetic sequence is when you can obtain the next term in the sequence by adding or subtracting some number from the previous term. We learned that a geometric sequence is similar, but the next term is obtained by multiplying the previous term by some number. We wrote out equations for each of these examples.
We then looked at series, saying that they were just sequences with all of the terms added up. We reviewed notation for infinite series including set notation and sigma notation. I advised her that most of the focus of what we will be doing with infinite series is determining if the series converges or diverges. We discussed that if as n approaches infinity the series approaches a number, then the limit exists, and the series converges. If instead, as the series approaches infinity, the limit does not exist, then the series diverges. We discussed that series that repeat themselves (2, 4, 2, 4, 2, 4, "_) do not converge, they diverge.
We then looked at several properties of infinite series, which approach limits including addition, multiplication by a constant, multiplication, and division. We continued by doing several examples and, in the process, reviewed how to calculate limits for polynomials with the larger degree in the numerator (no limit exists), the degree the same in the numerator and denominator (limit=ratio of the coefficients) and the larger degree in the denominator (limit=0). We reviewed the idea that if a "larger infinity"ù is in the denominator, that it's more important than a "lesser infinity"ù in the numerator, and the limit=0.
We did many example problems next, in which she was able to tell me the next item in a sequence. Then, we practiced writing down the equation for the series. In one example, the sequence was 1, (-3/2), 9/4, (-27/8), ?. We found that the following term was 81/16 and that the equation for the sequence could be represented as a(n)=(-3/2)^(n-1). We also did several problems in which we found the limit of the series and therefore concluded whether it converged or diverged. We reviewed that you can still use L'Hopital's Rule to calculate the limit when a direct substitution of n? infinity gives an undefined ratio of either infinity/infinity or 0/0. I suggested to the student that she works on her mental math, in her review of basic functions, their derivatives, and their integrals. We scheduled two more sessions so that we can adequately review these sections before her upcoming exam. Good class: 3/5 stars."