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e is irrational.

e is irrational.

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 4. By evaluating the Taylor series for the exponential function: at x = 1, we get the formula e = 1+ (a) Let Sn = X x² 1! xn + +.. + +... 2! n! 1 1 1 e = 1 + + + + 1! 2! 3! In this problem, you will prove that e is irrational. + 0≤e-Sn ≤ 1 n! n 1 Σ, the n-th partial sum of above series. Show that k!' k=0 . 1 1 n n! .... + (b) Assume e is rational, and say a/b is the reduced fraction representing e. Apply the previous result to n = b and arrive at a contradiction.
Transcribed Image Text:Problem 4. By evaluating the Taylor series for the exponential function: at x = 1, we get the formula e = 1+ (a) Let Sn = X x² 1! xn + +.. + +... 2! n! 1 1 1 e = 1 + + + + 1! 2! 3! In this problem, you will prove that e is irrational. + 0≤e-Sn ≤ 1 n! n 1 Σ, the n-th partial sum of above series. Show that k!' k=0 . 1 1 n n! .... + (b) Assume e is rational, and say a/b is the reduced fraction representing e. Apply the previous result to n = b and arrive at a contradiction.
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