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InvisibleLetto
Load Universeinto Cannon. Aimat Brain. Fire.
Registered: 12/13/02
Posts: 2,321
Calculus
    #2573949 - 04/18/04 02:56 PM (13 years, 2 months ago)

Since Shroomery doesn't have a math forum, I think this would be the best forum for me to ask for homework help. (I'm not asking for you to give me the work straight up, just any tips, so please don't flame me)

The assigned problem is Prove using L'Hopital's Rule that lim (n->infinity) (1 + 1/n)^n = e.

I first tried combining the term to get:

lim [(n+1)/n^2]^n = lim (n+1)^n/n^(2n)

but after taking the derivative (as per L'Hopital's Rule), I couldn't figure out anything, and enough of the terms didn't cancel.

So I tried to get the equality another way (not using L'Hopital's Rule) by setting

ln y = ln lim (n->infinity) (1 + 1/n)^n
ln y = n * lim ln (1 + 1/n) --- ln 1/n = 0, so it's just ln 1 = 0
ln y = n * 0 = 0, but this cannot exist.

Can somebody be kind enough to tell me where I'm going wrong or if there's a better method for me to use (preferably using L'Hopital's Rule)? Any help is definitely appreciated!


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Invisibleluvdemshrooms
Two inch dick..but it spins!?


Registered: 11/29/01
Posts: 34,203
Loc: Lost In Space
Re: Calculus [Re: Letto]
    #2573962 - 04/18/04 03:00 PM (13 years, 2 months ago)

Blue. The answer is blue.


--------------------
You cannot legislate the poor into prosperity by legislating the wealthy out of prosperity. What one person receives without working for another person must work for without receiving. The government cannot give to anybody anything that the government does not first take from somebody else. When half of the people get the idea that they do not have to work because the other half is going to take care of them and when the other half gets the idea that it does no good to work because somebody else is going to get what they work for that my dear friend is the beginning of the end of any nation. You cannot multiply wealth by dividing it. ~ Adrian Rogers


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OfflineMorbidHamster
Total Head Fook
Registered: 10/21/02
Posts: 121
Loc: Un-United Kingdom
Last seen: 12 years, 1 month
Re: Calculus [Re: Letto]
    #2574225 - 04/18/04 04:10 PM (13 years, 2 months ago)

I think my brain just melted  :crazy2:


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InvisiblePhencyclidine
Molecule

Registered: 06/02/03
Posts: 2,915
Re: Calculus [Re: Letto]
    #2581287 - 04/20/04 03:41 AM (13 years, 2 months ago)

Found it. I'll use x instead of n. inf. = "infinity"

[lim x to inf.] (1 + 1/x)^x
We have to change this to the form 0 / 0 or inf. / inf.
let y = (1 + 1/x)^x
thus [lim x to inf.] ln y = [lim x to inf.] x ln (1 + 1/x)
divide by (1/x) / (1/x)
so [lim x to inf.] ln y = [lim x to inf.] < ln (1 + 1/x) > / ( 1 / x)
which is of the form 0 / 0
apply L'Hopital's theorem
[lim x to inf.] ln y = [lim x to inf.] < 1 / (1 + 1/x) * ( -1 / x^2 ) > / ( -1 / x^2)
the ( -1 / x^2) terms cancel out, leaving
[lim x to inf.] ln y = [lim x to inf.] 1 / (1 + 1/x)
[lim x to inf.] e^(ln y) = [lim x to inf.] e ^< 1 / (1 + 1/x) >
[lim x to inf.] y = e^1

since y = (1 + 1/x)^x
[lim x to inf.] (1 + 1/x)^x = e


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