exponential function (1 Viewer)

[Constip8edSkunk]

I am having a bit of a mind block here and need some help.

I think this is covered in 4U:

given that (1 + 1/k)^k < e < (1 + k)^k+1 for k E Z+ ,
show that (1 + n)^n / e^n < n! < (1 + n)^(1+n) / e^n where n E Z+

 

[Mill]

Seems similar to Question 7 of the 4u HSC in 2002 off the top of my head. I haven't looked at in 18 months though, so that's a bit of a guess.

 

[Affinity]

hey you are doing the same course as me

skunk, follow the hints,

e>(1+1)
e>(1+(1/2))^2
...
e>(1+1/n)^n

we can rewrite as

e > (2/1)
e> (3/2)^2
..
e>((n+1)/n)^n

multiply the inequalities together

notice the products

(2/1)(3/2)...((n+1)/n) 'telescopes' and gives (1+n)

(3/2)(4/3)... = (1+n)/2

...

so
e^n > ([(1+n)^n]/n!

similarly

(2/1)^2 > e

(3/2)^3 > e

...

multiply

gives (n+1)^(n+1)/n! >e^2

 

[Grey Council]

should this be in the 'extracurricular' maths forum?

and crappo your good Affinity. I think i've said this before somewhere.

 

[Constip8edSkunk]

ah i see... thanks mate!