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Proving that e is Irrational In this project, we use the Maclaurin polynomials for e to prove that e is irrational. The proof relies on supposing that e is rational and arriving at a contradiction. Therefore, in the following steps, we suppose e = ris for some integers r and s where s + 0. 1. Write the Maclaurin polynomials po(x),P1(?). P2(x). P3(x). P4(x) for e*. Evaluate po(1).P1(1). P2(1). P3(1).P4(1) to estimate e. 2. Let R,(x) denote the remainder when using p,(x) to estimate e. Therefore, R,(x) = eš – p„(x), and R,(1) = e - P„(1). Assuming that e =- - for integers r and s, evaluate R,(1), R¡(1), R2(1), R3(1), R4(1).

Proving that e is Irrational In this project, we use the Maclaurin polynomials for e to prove that e is irrational. The proof relies on supposing that e is rational and arriving at a contradiction. Therefore, in the following steps, we suppose e = ris for some integers r and s where s + 0. 1. Write the Maclaurin polynomials po(x),P1(?). P2(x). P3(x). P4(x) for e*. Evaluate po(1).P1(1). P2(1). P3(1).P4(1) to estimate e. 2. Let R,(x) denote the remainder when using p,(x) to estimate e. Therefore, R,(x) = eš – p„(x), and R,(1) = e - P„(1). Assuming that e =- - for integers r and s, evaluate R,(1), R¡(1), R2(1), R3(1), R4(1).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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I am in no rush to get this done but, if anyone can help with this entire block, I would appreciate it greatly. Thank you so much!

STUDENT PROJECT Proving that e is Irrational In this project, we use the Maclaurin polynomials for e to prove that e is irrational. The proof relies on supposing that e is rational and arriving at a contradiction. Therefore, in the following steps, we suppose e = ris for some integers r and s where s + 0. 1. Write the Maclaurin polynomials po(x), p1(x), P2(x), P3(x), P4(x) for e. Evaluate po(1), P1(1). P2(1), P3(1), P4(1) to estimate e. 2. Let R,(x) denote the remainder when using p„(x) to estimate e. Therefore, R,(x) = e* – P„(x), and R,(1) = e - P„(1). Assuming that e = - for integers r and s, evaluate R,(1), R;(1), R,(1), R3(1), R4(1). 3. Using the results from part 2, show that for each remainder Ro(1), R1(1), R2(1), R3(1), R4(1), we can find an integer k such that kR,(1) is an integer for n = 0, 1, 2, 3, 4. 4. Write down the formula for the nth Maclaurin polynomial p,(x) for e* and the corresponding remainder R,(x). Show that sn!R,(1) is an integer. 5. Use Taylor's theorem to write down an explicit formula for R„(1). Conclude that R,(1) # 0, and therefore, sn!R,(1) # 0. 6. Use Taylor's theorem to find an estimate on R,(1). Use this estimate combined with the result from part 5 to show se that sn!R„(1) <; Conclude that if n is large enough, then sn!R,(1) < 1. Therefore, sn!R,(1) is an integer with n+1* magnitude less than 1. Thus, sn!R,(1) = 0. But from part 5, we know that sn!R,(1) # 0. We have arrived at a contradiction, and consequently, the original supposition that e is rational must be false.
Transcribed Image Text:STUDENT PROJECT Proving that e is Irrational In this project, we use the Maclaurin polynomials for e to prove that e is irrational. The proof relies on supposing that e is rational and arriving at a contradiction. Therefore, in the following steps, we suppose e = ris for some integers r and s where s + 0. 1. Write the Maclaurin polynomials po(x), p1(x), P2(x), P3(x), P4(x) for e. Evaluate po(1), P1(1). P2(1), P3(1), P4(1) to estimate e. 2. Let R,(x) denote the remainder when using p„(x) to estimate e. Therefore, R,(x) = e* – P„(x), and R,(1) = e - P„(1). Assuming that e = - for integers r and s, evaluate R,(1), R;(1), R,(1), R3(1), R4(1). 3. Using the results from part 2, show that for each remainder Ro(1), R1(1), R2(1), R3(1), R4(1), we can find an integer k such that kR,(1) is an integer for n = 0, 1, 2, 3, 4. 4. Write down the formula for the nth Maclaurin polynomial p,(x) for e* and the corresponding remainder R,(x). Show that sn!R,(1) is an integer. 5. Use Taylor's theorem to write down an explicit formula for R„(1). Conclude that R,(1) # 0, and therefore, sn!R,(1) # 0. 6. Use Taylor's theorem to find an estimate on R,(1). Use this estimate combined with the result from part 5 to show se that sn!R„(1) <; Conclude that if n is large enough, then sn!R,(1) < 1. Therefore, sn!R,(1) is an integer with n+1* magnitude less than 1. Thus, sn!R,(1) = 0. But from part 5, we know that sn!R,(1) # 0. We have arrived at a contradiction, and consequently, the original supposition that e is rational must be false.
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