(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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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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