Define 0! = 1. For any positive integer n, define n! = n- (n- 1)!. For example, 6! = 6x5x4x3x2×1= 720. 4" Use induction to show that (2n)! for all integer n> 2. n+1 (n!)2
Define 0! = 1. For any positive integer n, define n! = n- (n- 1)!. For example, 6! = 6x5x4x3x2×1= 720. 4" Use induction to show that (2n)! for all integer n> 2. n+1 (n!)2
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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![1. Define \( 0! = 1 \). For any positive integer \( n \), define \( n! = n \cdot (n-1)! \). For example, \( 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \).
Use induction to show that
\[
\frac{4^n}{n+1} < \frac{(2n)!}{(n!)^2}
\]
for all integer \( n \geq 2 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faad9e4ed-ed50-4fe3-96ca-72ed3d364f6f%2Fb12710ea-e41f-429e-a99f-9005dd0b235b%2F0lenov4_processed.png&w=3840&q=75)
Transcribed Image Text:1. Define \( 0! = 1 \). For any positive integer \( n \), define \( n! = n \cdot (n-1)! \). For example, \( 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \).
Use induction to show that
\[
\frac{4^n}{n+1} < \frac{(2n)!}{(n!)^2}
\]
for all integer \( n \geq 2 \).
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