2. Zino's dichotomy paradox is a practical depiction of a limit at infinity. The concept is that if you break down a journey into walking half way to an area consecutively, the distance will split infinitely. The function we end up with is _ ] (? )". If we take this equation and add (1/2 + 1/4 + 1/8....) it would become infinitely smaller and we would never reach what we are trying to reach. However the function is a geometric series that is convergent to 1. This practical example further explains last weeks example of how 0.9999...=1. With a value getting recursively smaller there cannot be any value between the point where infinite recursion happens and the limit value. 3. Grandi's series an alternating series which is follows the pattern (1-1+1-1+1....). The series is divergent and can either solve to 1 or 0 depending on where you place the brackets (f you treat it as a telescoping series). An interesting point of the history of the series is if you assign the series as a geometric series and assign it to a variable you obtain a third value [S = 1 - 1 + 1 -1 + 1 -1 .....][1 - S =1 -(1-1+1-1+1. ..) = S'] thus [1 - S = S] [1 = 2S] S = .A debate over which is the "correct answer" occurred for over 100 years until the 19th century until the practice of using the average of partial sums became widely accepted. While Grandi's series cannot have a limit because it never gets closer to one number or the either, by taking the average of partial sums, you can achieve a repeatable solution from this type of indeterminate series