en+1 – 1 e(n+1)z en+1 – 1 (n+ 1) · (n+ 1)! dr < 33- (n+ 1) · (n+ 1)! (n + 1)!

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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In this task we will investigate how we can estimate integrals we do not have a chance at by the usual rules by using Taylor's formula. You can use without justification that 2 <e <3 and that the function given by g (x) = e^x is increasing. Justify that for all real numbers x and positive integers n prove that the e^e^x expansion (see picture) is true for a Sx ∈ (0, e^x ). Then use what you know to explain why this is correct (see second picture)

e(n+1)x
ee" = 1+e" +
2
enr
+ esz
n!
+
6
(n + 1)!
Transcribed Image Text:e(n+1)x ee" = 1+e" + 2 enr + esz n! + 6 (n + 1)!
en+1 – 1
e(n+1)x
en+1 – 1
(n+ 1) · (n+ 1)!
(n + 1)!
dr < 33-
(n + 1) · (n+ 1)!´
0.
Transcribed Image Text:en+1 – 1 e(n+1)x en+1 – 1 (n+ 1) · (n+ 1)! (n + 1)! dr < 33- (n + 1) · (n+ 1)!´ 0.
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