Is Bernoulli's Inequality Applicable to Prove These Inequalities?

[lep11]

Homework Statement

Prove that
a.) (1-(1/n²))ⁿ > 1- 1/n

b.) (1+ 1/(n-1))^n-1 < (1 + 1/n)ⁿ

when n=2,3,4,5,...

Homework Equations

[/B]
Bernoulli's inequality
(1+x)ⁿ ≥ 1+nx,
when x ≥-1 and n=2,3,4,5,...

(1+x)ⁿ >1+nx,
when x ≥-1, x≠0 and n=2,3,4,5,..

The Attempt at a Solution

a.)[/B] I applied Bernoulli's inequality.

First I checked 'the requirements'.
-1/n² > -1 because n=1,2,3,... and -1/n² ≠ 0 OK

Then (1-(1/n²))ⁿ > 1+ (- 1/n²)*n=1- 1/n Ok, done.

b.) I think I am supposed to apply Bernoulli's inequality as in part a, but don't have an idea how to get started.

 

[SammyS]

Homework Statement

Prove that
a.) (1-(1/n²))ⁿ > 1- 1/n

b.) (1+ 1/(n-1))^n-1 < (1 + 1/n)ⁿ

when n=2,3,4,5,...

Homework Equations

[/B]
Bernoulli's inequality
(1+x)ⁿ ≥ 1+nx,
when x ≥-1 and n=2,3,4,5,...

(1+x)ⁿ >1+nx,
when x ≥-1, x≠0 and n=2,3,4,5,..

The Attempt at a Solution

a.)[/B] I applied Bernoulli's inequality.

First I checked 'the requirements'.
-1/n² > -1 because n=1,2,3,... and -1/n² ≠ 0 OK

Then (1-(1/n²))ⁿ > 1+ (- 1/n²)*n=1- 1/n Ok, done.

b.) I think I am supposed to apply Bernoulli's inequality as in part a, but don't have an idea how to get started.

What do you know about $\displaystyle\ \frac{1}{n-1} \ ?$

 

[lep11]

What do you know about $\displaystyle\ \frac{1}{n-1} \ ?$

Could you please elaborate?

 

[SammyS]

Could you please elaborate?

Is it positive?

Does it have an upper bound ?

 

[lep11]

Is it positive?

Does it have an upper bound ?

It is always positive.
0 < 1/(n-1) ≤ 1
And it does have an upper bound.
What's the next step?

 

[Ray Vickson]

It is always positive.
0 < 1/(n-1) ≤ 1
And it does have an upper bound.
What's the next step?

We are not allowed to tell you that; PF rules require you to do the work. However, if you get stuck at some point, we are permitted to give hints, but first you need to reach the point of getting stuck on your own.

 

[lep11]

We are not allowed to tell you that; PF rules require you to do the work. However, if you get stuck at some point, we are permitted to give hints, but first you need to reach the point of getting stuck on your own.

I have read the forum rules. I did the work at part a and now I am stuck at part b.

0 < 1/(n-1) ≤ 1 But how that will help?

 

[lep11]

Anyone?

 

[lep11]

Nevermind. Now I figured it out.