This activity is designed to help your Pre-Calculus Honors or College Algebra students evaluate sequences and series in an end-unit review for Discrete Mathematics. There are 24 task cards in the activity. Students will find recursive and explicit forms of sequences, find the sum of finite and infinite series, determine convergent and divergent series, find nth terms, partial sums, P(K+1) term for induction proofs, and more.
An infinite series is a sum of an infinite number of terms. Of course, the indexing can start at any integer, but by the most common starting indices are 0 0 and 1 1 . Regarding the second summation notation, of course there is no "infinity-th" term, as infinity is an not an integer; however, the notation is a convenient way for us to say that we take the summation over all natural numbers. ... See more at expii.
Riemann Zeta Function https://en.wikipedia.org/wiki/Riemann_zeta_function It is a function of a complex variable s that analytically continues the sum of the infinite series which converges when the real part of s is greater than 1. More general representations of ζ(s) for all s are given below (see at the wikipedia page of description). The Riemann Zeta Function plays a pivotal role in Analytic Number Theory and has applications in physics, probability theory, and applied statistics.
Mechanical PI is a computing machine (re)placing the repetitive algorithm back into a physical, mechanical language. A constant rotation, pressing and repeating the calculator’s keys, approaching the number Pi, yet never reaching it. The machine utilizes the Leibniz formula for pi which is an infinite series of additions and subtractions of quotients. Each subsequent denominator in this series is the sum of the previous one plus two, starting with the value one.