Show that this means & must be positive. Finally show that of est. Question 5. (5) Suppose a is a natural number. Argue why it must be the case that. ( ex )" = exexex ex = e hx ~ time Hint : you know this is true from Question 2 if you replace h by a you get exex exta How is it true for cases where n= 3 0 / anger? Question 6 ( 6 ). can you prove that, forany natural numbern, en = (02 ) ? " Hint : Start with e" = ex" = ( en ) where The last step comes from question s question 7. ( 7) positive . Suppose s = m is any rational number because in and n are natural Show that e= ( en ) .. = (9) Note : there is no proof that efx = ( ex ) if tis an irrational number like TI or N2. But, there is a result that states any irrational number to can be expressed as the limit of a sequence of ration numbers that converge to t. Such sequences alwa exist . ( think about the decimal expansion of an irration number). So, The custom is to think of an irrational as a limit of irrationals and say that what is true for rationals to also true for irrational .Due November 16, 2022 Econ 1530 Assignment # 2. Goal. This assignment is about the exponential function that is written e" or sometimes exp ( 2 ). The exponential function, as we will learn, is uniquely related to the logarithmic function, In x by inversion ( a notion we will learn soon). All of the properties of, e" can be deduced from its definition and some careful reasoning . This is what you will learn in this assignment . Note: This assignment is difficult. Use any source of help you want. O. Definition. The exponential function is defined as the Sum of an infinite polynomial function. In particular : ( D) 2"= 1 + 26 + 26 4 20 + 20 + ... ( without and) 2! 3! 4! where n! = n * ( n - 1) * ( 1 - 2 ) * ..$ 1 in called n " factorial " and it is the product of the natural numbers from n down to 1 By convention, O! is taken to be 1. ex t called a polynomial because it is the sum of many monomials like 1, 2, 2, * ..., all with a coefficient term ( 1 , 1 , 2 1 = 2, 3 1 = 6 .. ) multiplying it. 1. Question 1. Does e" converge to a finite number if 2 is a finite number ? That is , is the value that ex provides finite if 2 is finite ? It turns out that the answer is you. But we need to approach the answer in stages .2. ( 2 . ) . can't . Recall that we established geometrically that the AllM : first form of S. 1 + ( 1 ) + ( 2 ) + ( ) + - suing first 2 terms - sum 7first 31 terms limited to the value 2. It never exceeded 2 1 2 so it was bounded by 2 and hence finite. each extra tesme takes you Now look at es half way to 2 from * ..... your last location . 2! 3! Compare e? term by term with S and conclude that ez