kk, u made me open the damn book, I'm not going to put the effort in to solve all the steps but I'll give you what my old book has. It basically has the same question but gives you clues to solving.

an = (1 + 1/n)^n
First (a) it wants you to show that if 0<=a<b
[b^(n+1)-a^(n+1)]/b-a < (n+1)b^n
then, (b)
deduce that b^n[(n+1)a-nb] < a^(n+1)
then, (c)
use a = 1+1/(n+1)
b = 1+1/n
in part (b) to show that {an} is increasing

I hope this helps. Basically what you want is part c but it looks like you need to do part a,b for it to make sense.

ProudToBeANoob

Thailand54 Posts

September 22 2003 20:56 GMT

#182

just a remark like that, but is it not useful to take the exp formula of (1+1/x)^x, and using the fact that both log and exp are increasing functions ? I don't remember well all the conditions to say that f(something increasing) with f increasing is also increasing (i think that everything should be strictly positive which is the case), maybe i am totally wrong...

LpThaPro

Canada21 Posts

September 22 2003 22:06 GMT

#183

Hey Smart Kid 'BigBalls', Since you are a Math major, I think you should be more respectful and thankful to France as it is THE country which contributed the most in the history of mathematics.

Oh yeah, by the way, France is THE most visited country in the World and by far.
I think you should go there and realize that there are far more important things in life than military victories.

-Peace

ProudToBeANoob

Thailand54 Posts

September 22 2003 22:13 GMT

#184

I totally agree, but please, don't send more people in France to visit, they will discover how badly we welcome tourists :p

Anyway, fed up with this 'gueguerres' usa/france and antiUSA/proUSA, it doesn't lead anywhere and just enforces stupidity where it already stands strong...

Commander{+}

United States2878 Posts

September 22 2003 22:17 GMT

#185

On September 23 2003 04:17 BigBalls wrote:

Show nested quote +

i know its e, but i have to prove it is strictly increasing, thats the point of the exercise

.

i did cross multiply. i got (n+1)^2n+1 <= n^n * (n+2)^n+1 its not intuitively obvious that one is larger than the other. I tried induction and it wasnt working out (assume this is true and then show that (n+2)^2n+3 <= n+1^n+1 * (n+3)^n+2 I couldnt get it to show this last step.

Tutor me... bitch!

C)R

Canada549 Posts

September 22 2003 22:47 GMT

#186

On September 23 2003 05:56 ProudToBeANoob wrote:
just a remark like that, but is it not useful to take the exp formula of (1+1/x)^x, and using the fact that both log and exp are increasing functions ? I don't remember well all the conditions to say that f(something increasing) with f increasing is also increasing (i think that everything should be strictly positive which is the case), maybe i am totally wrong...

this isn't a function, it's an infinite series, you can only prove it's increasing (by definition) by showing an<=an+1 for all n>=1

X)Benny

France1270 Posts

September 23 2003 01:58 GMT

#187

LeJester,
you've got a point. So I'll put it this way.
Saying that "any food in the world is better than US food" is just my opinion, you don't have to take it. Well at least this opinion is based on experience: I've spent 5 months in the US, 6 in Canada, 1 in Bolivia, 1 in Japan... As for Blackjack and snails, I guess he never tried them, I guess he has never been to France either, so how can he claim bigmac>snails?

BigBalls,
Sorry I left work just when you released your problem. I'm glad we got back to more serious talks.

An=(n+1/n)^n
An+1=(n+2/n+1)^(n+1)
An+1/An=(n+2/n+1)*(n+2/n)^n > 1

gg

X)Benny

France1270 Posts

September 23 2003 02:02 GMT

#188

oups.... sorry, there's a mistake in the calculation, mornings don't help me think.

let me think more...

ProudToBeANoob

Thailand54 Posts

September 23 2003 02:16 GMT

#189

On September 23 2003 07:47 C)R wrote:

Show nested quote +

this isn't a function, it's an infinite series, you can only prove it's increasing (by definition) by showing an<=an+1 for all n>=1

So what? If it works for every x in IR+*, it works for every n in IN*.
Proving f(a)<f(b) if a<b => an=f(n)<an+1=f(n+1) for n>=1.
But what i doubt about my remark is just the condition to say that fog(x) is increasing when g is increasing, it's a long time ago, i forgot it...

Anyway Benny's try seems a good way...Or we can try 'par recurrence' (i don't know the word in english).

X)Benny

France1270 Posts

September 23 2003 02:39 GMT

#190

oki;
it's not very smart but it works:
let f(x)=(1+1/x)^x , we want to show f increases in R+
f'(x)=[ln(1+1/x) - 1/1+x]*exp(...), we want to show f'>0
g(x)=f'/exp(...), we want to show g>0
g'(x)=1/(1+x)^2 - 1/x*(1+x) < 0
so g decreases. Since we also have lim(x->+oo) g = 0, it means g>0.

gg

ProudToBeANoob

Thailand54 Posts

September 23 2003 03:01 GMT

#191

hola je pensais pas qu'il fallait deriver etc...Y a pas juste un theoreme qui dit si g croissante et strictement positive (ici xlog(1+1/x) qui l'est si x>0), et si f est croissante (exp l'est par definition), alors fog (la composee de g par f) est croissante ? Peut-etre qu'il faut plus que la stricte positivite (de plus, je vois meme pas pourquoi l'imposer, donc chuis ptet a la rue)...

Sorry for non-speaking french, i just ask :p

hard to come back to 8 years ago...

Liquid`Drone

Norway28665 Posts

September 23 2003 03:49 GMT

#192

:E

shasta, if someone made a funny joke about norway, I'd laugh at it.
if someone made a funny joke about my mother, I'd laugh. if someone makes a funny joke about me, I laugh.

to everyone from north america who has said "the world hates the french because" or anything similar to that, you forget something. the world doesn't hate the french. europeans don't hate the french. the world hates USA. and for plenty of more better reasons than anyone has for hating the french!

but hating based on nationality is pretty fucking retarded in the first place. : )

likewise, everyone who are "proud" to be of any nationality are a bunch of fucking morons.

X)Benny

France1270 Posts

September 23 2003 04:28 GMT

#193

This old post is unavailable due to an encoding issue. Please contact an admin if you would like this post restored for historical reasons.

BinGBonG[gamei]

Netherlands514 Posts

September 23 2003 05:55 GMT

#194

HOLLAND !!! HOLLAND!! HOLLAND!!! hehe idiots =P

BigBalls

United States5354 Posts

September 23 2003 08:43 GMT

#195

On September 23 2003 11:39 X)Benny wrote:
oki;
it's not very smart but it works:
let f(x)=(1+1/x)^x , we want to show f increases in R+
f'(x)=[ln(1+1/x) - 1/1+x]*exp(...), we want to show f'>0
g(x)=f'/exp(...), we want to show g>0
g'(x)=1/(1+x)^2 - 1/x*(1+x) < 0
so g decreases. Since we also have lim(x->+oo) g = 0, it means g>0.

gg

cant use that. i used ln in one of my epsilon proofs and he said we werent allowed to use it yet.

i think im onto something. it has to do with the binomial expansion of n and n+1.

the n+1 expansion has a term that dominates everything in the n expansion. i think this is the best way to go about it.

gg :o

Mynock

4492 Posts

September 23 2003 09:52 GMT

#196

On September 23 2003 12:49 Liquid`Drone wrote:
:E

shasta, if someone made a funny joke about norway, I'd laugh at it.
if someone made a funny joke about my mother, I'd laugh. if someone makes a funny joke about me, I laugh.

to everyone from north america who has said "the world hates the french because" or anything similar to that, you forget something. the world doesn't hate the french. europeans don't hate the french. the world hates USA. and for plenty of more better reasons than anyone has for hating the french!

but hating based on nationality is pretty fucking retarded in the first place. : )

likewise, everyone who are "proud" to be of any nationality are a bunch of fucking morons.

Humm Drone... pretty nice.

-Mynock

KiDDy

Finland12 Posts

September 23 2003 10:01 GMT

#197

drone seems like a wise guy... nice.

my opinion about USA

and

so damn selfcentered country.

i'm pretty neutral about the french.

iD.GioM

France99 Posts

September 23 2003 10:43 GMT

#198

Drone is sooooo sexy !
Blackjack is retarded
man I told u I've experienced it ?
And u havent been in war with france have u ? so stfu ? thx !

ayer

Mexico164 Posts

September 23 2003 11:07 GMT

#199

On September 22 2003 23:22 BigBalls wrote:

Show nested quote +

you albanian?

It doesn't matter the whole point of this thread for us to realize we are all human and have buttsex with each other!

BlackJack

United States10495 Posts

September 23 2003 11:52 GMT

#200

On September 23 2003 19:43 iD.GioM wrote:
Drone is sooooo sexy !
Blackjack is retarded
man I told u I've experienced it ?
And u havent been in war with france have u ? so stfu ? thx !

I have no idea what you are talking about