(The number is irrational) Problem 6 above shows how to prov

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(The number is irrational) Problem 6 above shows how to prove that e is irrational by assuming the contrary and deducing

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(The number is irrational) Problem 6 above shows how to prove that e is irrational by assuming the contrary and deducing a contradiction. In this problem you will show that is also irrational. The proof for is also by contradiction but is rather more complicated, so it will be broken down into several parts. (a) Let f .x/ be a polynomial, and let g.x/ D f .x/ ? f 00.x/ C f .4/.x/ ? f .6/.x/ C D X1 jD0 .?1/j f .2j /.x/: (Since f is a polynomial, all but a finite number of terms in the above sum are identically zero, so there are no convergence problems.) Verify that d dx g0 .x/sin x ? g.x/ cos x D f .x/sin x; and hence that Z 0 f .x/sin x dx D g. / C g.0/. (b) Suppose that is rational, say, D m=n, where m and n are positive integers. You will show that this leads to a contradiction and thus cannot be true. Choose a positive integer k such that . m/k=k

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