Understanding the First Part of an Inequality Factorial

Thread starter Unassuming
Start date Oct 21, 2008
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Factorial Inequality

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SUMMARY

The inequality involving factorials is established as follows: the expression 1/(n+1)! multiplied by the geometric series 1 + 1/(n+1) + 1/(n+1)² + ... + 1/(n+1)ᵏ is less than 1/(n!n) and also less than 1/n. By defining a as 1/(n+1), the series simplifies to a recognizable geometric series, allowing for straightforward calculation and verification of the inequality's validity. This mathematical relationship is crucial for understanding factorial growth and convergence in series.

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Understanding of factorial notation and properties
Familiarity with geometric series and their sums
Basic knowledge of inequalities in mathematics
Ability to manipulate algebraic expressions

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Study the properties of geometric series and their applications
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Mathematicians, educators, students in advanced mathematics, and anyone interested in the properties of inequalities and factorials.

Oct 21, 2008

Unassuming

Why is the first part of this inequality true?

1/(n+1)! [ (1 +1/(n+1) +1/(n+1)^{2} +...+ 1/(n+1)^{k} ]
< 1/(n!n) < 1/n

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Oct 21, 2008

HallsofIvy

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Homework Helper

Let a= 1/(n+1) and that sum becomes 1+ a+ a^2+ ...+ a^k, a geometric series. You can write down a simple for for it. Once you have simplified that, it should be clear.