Assume e is rational and e = ², where p and q are integers and q ± 0. (a) Start with the Taylor Series expansion of e*. You do not have to find it yourself, it's in your class-notes. Then write down a series representing e. (b) Consider the number M= q!| e- i=0 Explain why M must be a positive integer (i.e. a whole number, not a fraction). (c) Show that M can be simplified to 00 1 M = q! ) i=q+1

Assume e is rational and e = ², where p and q are integers and q ± 0. (a) Start with the Taylor Series expansion of e*. You do not have to find it yourself, it's in your class-notes. Then write down a series representing e. (b) Consider the number M= q!| e- i=0 Explain why M must be a positive integer (i.e. a whole number, not a fraction). (c) Show that M can be simplified to 00 1 M = q! ) i=q+1

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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Question

Assume e is rational and e = 2, where p and q are integers and q ± 0.
(a) Start with the Taylor Series expansion of e*. You do not have to find it yourself, it's in your class-notes.
Then write down a series representing e.
(b) Consider the number
M= q!|e -
i=0
Explain why M must be a positive integer (i.e. a whole number, not a fraction).
(c) Show that M can be simplified to
1
M = q! )
i!
i=q+1
*This happened in Ancient Greece.

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