Question

I am learning how to find the limit of a sequence and this is the problem. (n!)/(n^n). So the first step is to expand the numerator, and in the video the numerator expands as follows: (n) (n-1) (n-2) .....(2)(1). I and confused about how the person carried out the expansion. how do they get (2) (1) as the last terms? and if I did the expansion, wouldn't it just continue to (n-1)(n-2)(n-3) (n-4)?

Answer 1

I am*

Answer 2

hmm....n! means 2*3*4*.....*(n-2)*(n-1)*(n) right?

Answer 3

if n=3, to keep things simple: we got:

(3)(3-1)(3-2) if we go any further we end up wit a (3-3)=0 and that messes up the factorial right?

Answer 4

yes. but in this case we dont know what n is. so I thought it would just go onto infinity. how did they get (2)(1) at the end?

Answer 5

n doesnt mean infinity it just means "a number that we can choose" n=3 is perfectly legit.

the logic is that when you begin with a number, lets call it "n" that the factorial of the number will be:

(n)(n-1)(n-2)(n-3) .... all the way down to.....(n-n+3)(n-n+2)(n-n+1)

n-n+3 = 3 right?

n-n+2 = 2 right? you see the logic?

Answer 6

regardless of the value of "n"; when it gets close to 0 as it moves along down the number line

Answer 7

kk let me read your reply. one minute..

Answer 8

okay I think I am seeing the logic. please continue, when n gets close to 0....

Answer 9

that was a typo...that when wasnt supposed to be there :)

we can start at "1" and move all the way to "n" and it will be the same value right? so lets turn this factorial around and start with 1...............n

(1)(2)(3)(4) ... (n-3)(n-2)(n-1)(n) do you see how gettting close to "n" is just (n-3) and those other notations?

Answer 10

think of it like this: you really want to get to the fridge to get a soda.... but someplace between here and the fridge, you stop.

look behind you and you will see how far you have come: (1step)(2steps)(3steps).

now look ahead of you towards the fridge, how far to you have to go?:

(fridge-3steps)(fridge-2steps)(fridge-1steps)(fridge,your there)

Answer 11

oh wow!!!!! great analogy! I am trying to apply that to the n!. so I get to the fridge (n) and want to retrace my steps. it would be (n-1) (n-2) (n-3) (n-4)........

Answer 12

yes..... exactly

Answer 13

your between the fridge and the sofa after getting your soda, the fridge behind you is:

(n)(n-1)(n-2)(n-3)(n-4) ...you are here now..(3)(2)(1, safely back at the sofa)

Answer 14

your just counting down the steps from your first trip.....

Answer 15

and is the midpoint I am standing at considered 0 on the number line? because the values become postive?

Answer 16

not really, we are measureing a distance, and distance is always a positive value; but if we were to use the number line it would not matter..... but in these exansions you really want to start at an origin of 0 and measure a distance of "n" steps.

Answer 17

les say we want to go 10 steps.....

1,2,3,4,5 were are here, 10-4,10-3,10-2,10-1,10.

10-4 = 6

10-3=7

10-2=8

10-1=9

10=10

Answer 18

Thank you so much for the thorough explanations with great examples. I think I am getting a grasp of this concept now. I really appreciate your help! :) :) :)

Answer 19

youre welcome...... math jargon can be rather complicated. these mathmatickers just aint got nothing better to do i think :)