natural base "e" explained! somewhat!

Discussion in 'Off Topic [BG]' started by Brad Barker, Nov 20, 2002.

  1. Brad Barker

    Brad Barker Supporting Member

    Apr 13, 2001
    berkeley, ca

    okay, that said...

    how the heck did euler decide that this x-value would equal [1/0! + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! + 1/6!...(ad infinitum)]?

    on that page, dheera explains the usage of an "x limit." perhaps if i understood just what the heck that is, i would have an idea.

    help, please? i'm so close!

    (btw, simply using those numbers that i listed in the bracket--sans ad infinitum--, you would be able to calculate e to the third decimal. the rest of those numbers are a whole lotta nothin', i tell ya!).
  2. Im pretty sure that it represents a curve, and that curve evens out around 2.71.

    However, I havent taken math in over 2 years, so I might be wrong.
  3. rustyshakelford


    Jul 9, 2002
    I had thought your sequence could be manipulated to yield e.

    Now, I don't think so.

    While I was at, they have e to 2 billion plus places.

    The derivation is here:

    Please see section 1.2

    I recall a lucid derivation in the classic Calculus Made Easy.

    Hope this helps.

  4. BertBert


    Nov 9, 2002
    Euler proved that the function f(x) = e^x can be written as a power series:

    e^x = 1 + x + x^2/2! + x^3/3! + x^4/4! + x^5/5! + ...

    This is proved in most standard calculus textbooks in the section on Taylor and MacLaurin series. Once you have shown that e^x has that power series expansion, just plug x = 1 in to both sides and you've got your infinite series for the number e. :cool:

    As for the limit on that web site, just take the expression to the right of the "lim x->0" sign -- (1+x)^(1/x) -- and plug in values of x that get closer and closer to, but not equal to, x = 0. Try x = 0.5, x=0.1, x=0.01, etc. Watch what the outputs start heading towards.

    I wear my math geekhood as a badge of honor!
  5. jazzbo


    Aug 25, 2000
    San Francisco, CA
    Hey Bert, you should check your numbers, I think you might be slightly off on a couple of things. To point:

    While the concept is true, don't forget subsect e subdivided by the incongruent formula 4x(3^ab) / e=Fb is going to yield a higher north rotation axis because of polar flux. The Taylor series has its biggest problems in its 3rd down percentage. Why they don't punt, I don't know.

    Good point, but "lim xy>4^666" is still going to reciprocate the flatulent fortitude myagnified by a diversent and recabituated x - y = 4a - b. People always forget that.

    The main point is, not to forget to subsifortimitigate each non-entity negative xyz rehabituationaly stacked factor.

    I hope that cleared that up.
  6. Jazzbo, please.

    Everybody knows that "rehabituationaly" is supposed to have TWO "l"s.

    Can't you even spell basic English? ;)
  7. jasonbraatz


    Oct 18, 2000
    Oakland, CA
    sweet jesus! i think my head just exploded. :eek:
  8. BertBert


    Nov 9, 2002
    I always wondered how that BCS ranking formula works. What a relief!

  9. Suddenly I'm not so sure that Jazzbo's theory lessons are aimed at my level.......;)
  10. jasonbraatz


    Oct 18, 2000
    Oakland, CA
    HAHAHA! the first time i read thru jazzbos post i just skimmed it, saw formulas, and assumed it was something relavant.

    then i read it again....


  11. Subtle, isn't he!:)
  12. rustyshakelford


    Jul 9, 2002
    e is the limit, as n approaches infinity, of:

    (1 + 1 / n)^n

    The binomial theorem says:

    (x + a)^n=Sum[(n!/(k!(n-k)!)) (x^k) (a^(n-k))]

    Note this is the product of three terms:
    #1 (n!/(k!(n-k)!))
    #2 (x^k)
    #3 (a^(n-k))

    With x=1 and a=(1/n)
    and with very large n,

    the second term is always 1
    the third term approaches 1 as n gets big
    so all your left with is the first term.

    And when you use infinity in the first term for n, n! and (n-k)! cancel out, so all you are left with is the sum of 1/k!
  13. Velkov


    Jan 17, 2001
    Lansdowne, Ontario
    I am with BertBert on this one. I haven't done any Calculus for a few years now but that looks like a Taylor series expansion.
  14. Brad Barker

    Brad Barker Supporting Member

    Apr 13, 2001
    berkeley, ca
    i have to admit, i didn't understand a DAMN THING on that page!

    what the hell is a derivative?

    what the hell is a limit?

    i'm just not getting it!
  15. Derivatives and limits are concepts one learns in a beginning Calculus course. What level of math are you currently taking?
  16. jazzbo


    Aug 25, 2000
    San Francisco, CA
    While it's true that you can spell rehabituationaly with two "L"s, it can also be spelled with one, just like the word "cancelled/canceled." Both are correct.

    Yes. How else do you think OSU would be ranked #2. True, they did struggle to that overtime win against Illinois, but as a Buckeye fan, I gotta think we should have them #1.

    I busted out the knowledge on ya. It can be hard to understand for some, but don't worry, I'll simplify stuff in the future for you.
  17. Brad Barker

    Brad Barker Supporting Member

    Apr 13, 2001
    berkeley, ca
    i'm in pre-cal.

    so far, we've been reviewing algebra 2 (we didn't review the binomial theorem! :eek: )

    next chapter: trigonometry! :cool:

    i may be biting off more than i can chew at the moment!
  18. rustyshakelford


    Jul 9, 2002
    You have had the binomial theorem, but you didn't know it.

    (a+b)^2 = ?

    Binomial means 2 numbers.

    This: (a+b)^2 is a case where you are squaring the sum if two numbers.

    If you wanted to solve (a+b)^15, this would be tedious.

    However, with the binomial theorem, you can solve this more quickly. In this case, (a+b)^15, n=15, versus the previous example, (a+b)^2, n=2.

    Again, I recommend Calculus Made Easy by Sylvanus Thompson. It was originally printed in 1910. It is available in paperback for cheap. It is so popular, it was just reissued in hard cover.

    It is a quick read.

    Take care,