![]() Use Taylor’s theorem to write down an explicit formula for R n ( 1 ).Show that s n ! R n ( 1 ) s n ! R n ( 1 ) is an integer. Write down the formula for the nth Maclaurin polynomial p n ( x ) p n ( x ) for e x and the corresponding remainder R n ( x ).Using the results from part 2, show that for each remainder R 0 ( 1 ), R 1 ( 1 ), R 2 ( 1 ), R 3 ( 1 ), R 4 ( 1 ), R 0 ( 1 ), R 1 ( 1 ), R 2 ( 1 ), R 3 ( 1 ), R 4 ( 1 ), we can find an integer k such that k R n ( 1 ) k R n ( 1 ) is an integer for n = 0, 1, 2, 3, 4.Assuming that e = r s e = r s for integers r and s, evaluate R 0 ( 1 ), R 1 ( 1 ), R 2 ( 1 ), R 3 ( 1 ), R 4 ( 1 ). Therefore, R n ( x ) = e x − p n ( x ), R n ( x ) = e x − p n ( x ), and R n ( 1 ) = e − p n ( 1 ). Let R n ( x ) R n ( x ) denote the remainder when using p n ( x ) p n ( x ) to estimate e x.Write the Maclaurin polynomials p 0 ( x ), p 1 ( x ), p 2 ( x ), p 3 ( x ), p 4 ( x ) p 0 ( x ), p 1 ( x ), p 2 ( x ), p 3 ( x ), p 4 ( x ) for e x. ![]() Therefore, in the following steps, we suppose e = r / s e = r / s for some integers r and s where s ≠ 0. The proof relies on supposing that e is rational and arriving at a contradiction. In this project, we use the Maclaurin polynomials for e x to prove that e is irrational.
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