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(a) Use induction to prove that for all x > 0 and all positive integers n x² + 2! e* > 1+x + + 3! n! Hint: Observe e = 1+ | e'dt > 1+ | 1dt > 1+x, and then [ e'dt > 1+ 72 | e'dt | (1+t)dt > 1+x + e = 1+ 2 (b) Use part(a) to show if n is a positive integer, then et > x" for all æ sufficiently large. Hint: this is equivalent to showing >1 for x sufficiently large.

(a) Use induction to prove that for all x > 0 and all positive integers n x² + 2! e* > 1+x + + 3! n! Hint: Observe e = 1+ | e'dt > 1+ | 1dt > 1+x, and then [ e'dt > 1+ 72 | e'dt | (1+t)dt > 1+x + e = 1+ 2 (b) Use part(a) to show if n is a positive integer, then et > x" for all æ sufficiently large. Hint: this is equivalent to showing >1 for x sufficiently large.

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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Problem 1. (a) Use induction to prove that for all a > 0 and all positive integers n x2 et >1+x + 2! + 3! x" + n! Hint: Observe et = 1+ et dt > 1+ 1dt >1+x, and then e = 1+ e*dt > 1+ (1+t)dt > 1+x + 2 (b) Use part (a) to show if n is a positive integer, then et > x" for all æ sufficiently large. Hint: this is equivalent to showing > 1 for x sufficiently large.
Transcribed Image Text:Problem 1. (a) Use induction to prove that for all a > 0 and all positive integers n x2 et >1+x + 2! + 3! x" + n! Hint: Observe et = 1+ et dt > 1+ 1dt >1+x, and then e = 1+ e*dt > 1+ (1+t)dt > 1+x + 2 (b) Use part (a) to show if n is a positive integer, then et > x" for all æ sufficiently large. Hint: this is equivalent to showing > 1 for x sufficiently large.
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