9780023606922, Chapter 2 1, Problem 38E

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Chapter 2.1, Problem 38E Step-by-step solution Step of 3 Consider the upper bound for n In other en Let us first prove that k! We know, n (n-k)!k! n! = n(n-1)...(n-k+1) k! k! = k! Thus, k! Step of 3 to prove that \ en k It would suffice to prove that: To this end, let us take logarithms to the base e, this inequality implies that: This inequality can be easily proved, using the following integral: dx= Also, we know that, And dx≤ k(Ink-1) Thus, Step of 3 From which clearly follows that: Which means 1,1 Also, we already proved that: That n! Thus, multiplying the two equations we obtain that,