Consider the Expression K^n/n!

[toughcookie723]

Consider the Expression K^n/n!, where k is a positive real number and n! is the product of whole numbers from 1 to n.


PART A
Prove: For any value of k, there is a value of n for which k^n/n! > k^(n+1)/ (n+1)!


(So the premise is then : if n+1>k, then k^n/n! > k^(n+1)/ (n+1)!)


By induction, if n=2, then


1^2/2*1> 1^3/3*2*1
If true for n=m, then true for n=m+1


k^m/m! > k^(m+1)/(m+1)!


(k/m+1)* k^m/m! > k^(m+1)/(m+1)!* (k/m+1)


k^(m+1)/(m+1)! > k^(m+1)/(m+1)!* k/m+1


k^(m+1)/(m+1)! > k^(m+1)/(m+1)!* k/(m+1+1-1)


k^(m+1)/(m+1)! > k^(m+1)/(m+1)!* k/(m+2-1)


(and here I am stuck. I know I am supposed to end up at k^(m+1)/(m+1)! > k^(m+2)/(m+2)!)


Part B


Let H= {k^n/n!: n is a whole number}. Prove: H has a largest element.


You may assume that any finite set of real numbers has a greatest term (but note that H is not a finite set). Hint: Part (a) is the first step in the induction proof. What’s the rest of the induction proof and what’s the induction proof proving?).


-This I am totally lost in. I understand that when it says any finite set of real numbers has a greatest term refers to the set of n numbers. And since there is a finite set of n numbers and it has a largest element then H must have the largest element? Really no clue where to start or where I should arrive at if the premise of the proof is part (a).




HELP, Please!!!

 

[Deleted member 4993]

Consider the Expression K^n/n!, where k is a positive real number and n! is the product of whole numbers from 1 to n.


PART A
Prove: For any value of k, there is a value of n for which k^n/n! > k^(n+1)/ (n+1)!


(So the premise is then : if n+1>k, then k^n/n! > k^(n+1)/ (n+1)!)


By induction, if n=2, then


1^2/2*1> 1^3/3*2*1
If true for n=m, then true for n=m+1


k^m/m! > k^(m+1)/(m+1)!


(k/m+1)* k^m/m! > k^(m+1)/(m+1)!* (k/m+1)


k^(m+1)/(m+1)! > k^(m+1)/(m+1)!* k/m+1


k^(m+1)/(m+1)! > k^(m+1)/(m+1)!* k/(m+1+1-1)


k^(m+1)/(m+1)! > k^(m+1)/(m+1)!* k/(m+2-1)


(and here I am stuck. I know I am supposed to end up at k^(m+1)/(m+1)! > k^(m+2)/(m+2)!)


Part B


Let H= {k^n/n!: n is a whole number}. Prove: H has a largest element.


You may assume that any finite set of real numbers has a greatest term (but note that H is not a finite set). Hint: Part (a) is the first step in the induction proof. What’s the rest of the induction proof and what’s the induction proof proving?).


-This I am totally lost in. I understand that when it says any finite set of real numbers has a greatest term refers to the set of n numbers. And since there is a finite set of n numbers and it has a largest element then H must have the largest element? Really no clue where to start or where I should arrive at if the premise of the proof is part (a).




HELP, Please!!!

Why invoke induction .... when deduction will do.

kⁿ⁺¹/(n+1)! = kⁿ/(n)! * k/(n+1)

Thus for k < n+1 → n > k-1 we will have:

kⁿ⁺¹/(n+1)! < kⁿ/(n)!

 

[toughcookie723]

Thanks for your reply! The reason I involved induction is because the professor wanted it done that way. But you're absolutely right!

 

[lookagain]

Consider the Expression k^n/n!, where k is a positive real number
and n! is the product of whole numbers from 1 to n.


PART A
Prove: For any value of k, there is a value of n for which k^n/n! > k^(n + 1)/(n + 1)!


(So the premise is then : if n + 1 > k, then k^n/n! > k^(n + 1)/(n + 1)!


By induction, if n = 2, then


1^2/(2*1) > 1^3/(3*2*1)
If true for n = m, then true for n = m + 1


k^m/m! > k^(m + 1)/(m + 1)!


k/(m + 1)*k^m/m! > k^(m + 1)/(m + 1)![k/(m + 1)]


k^(m + 1)/(m + 1)! > k^(m + 1)/(m + 1)![k/(m + 1)]


k^(m + 1)/(m + 1)! > k^(m + 1)/(m + 1)![k/(m + 1 + 1 - 1)]


k^(m + 1)/(m + 1)! > k^(m + 1)/(m + 1)![k/(m + 2 - 1)]


(and here I am stuck. I know I am supposed to end up at k^(m + 1)/(m + 1)! > k^(m + 2)/(m + 2)!

toughcookie723,

regardless of whether your process is correct, you must use grouping symbols in certain places,
as suggested above because of the Order of Operations.


Also, as typed by me above, \(\displaystyle space \ out\) your symbols/characters horizontally
for greater readability. You have most of it crammed together.