4.2: Summation
Massachusetts Institute of Technology: Srini Devadas and Eric Lehman's "Sums and Approximations" URL
Read this lecture. Take your time to understand the transition from one step to the next in the proofs. This can be time consuming, but it is rewarding. Don't let the topic of the examples - annuities, for example - distract you. The domains (e.g. annuities, Taylor series, etc.) are important, but so is the method they illustrate. The method is more general than the specific domain of the example.
The reading mentions induction as a way of proving some of the expressions and then continues to give insight into the source of the expression. We will cover induction in Subunit 4.5 of this course.
Read Section 3 on pages 20 - 30.
For sequences, given a summation ∑n > = k (f(n), we can expand it to the series, f(k), f(k) + f(k + 1), f(k) + f(k + 1) + f(k + 2), .... Given the series, f(k), f(k) + f(k + 1), f(k) + f(k + 1) + f(k + 2), f(k) + f(k + 1) + f(k + 2) + f(k + 3), ..., we can write it as ∑n >= k f(n).
Read this brief note on summation.