The number it arises in the ratio circumference}radius of a circle. Another number of great importance is e = 2.71828- - - . In this question we will see a way this number arises. Consider $0 dollar invested at an annual interest rate of 100% [a lot!) that is compounded n E N {n 1':- 0) times per year. After the rst compounding period C grows to CU + lf-n). after two compounding periods to C (1 + 1 [11)2. after u. compounding periods (one year) to 3,, = c(1+ 1) 5 Ce.\" Tl. This denes the sequence en. The limit n > 30 is the limit of continuous compounding. {a} Calculate the rst ten values e1. e2. - - -e{ll]]. and plot them. (b) Using the AG inequality or otherwise. show that en+1 :2- en. 1 n+1 {c} To show that en is bounded above. consider the sequence fn = (1 + a) . Show numericallyr that {In} is a decreasing sequence. and calculate lim (fa c\"). Use this to \"l'x conclude that en is bounded above and the limit of both sequences is the number e. n Consider the exponential function f(x) = e, with e as defined in Q1: e = lim 1 n-+00 n In this question you will rigorously calculate f'(x). (a) Using the definition of the derivative, show that f'(x) = et lim ho h (b) Using the sequences {en} and {fn} of Q1, show that for all n > 1, ( 1+ 4 ) "