1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Basic Derivation with negative exponents

  1. Jul 9, 2009 #1
    Hello reader

    So I'm trying to teach mysel calculus with Thompson's Calculus made easy.
    I'm still in the very early stages and I got stuck with negative exponents.
    I'd basically like an explanation fof what's going on here.

    http://www.flickr.com/photos/26906498@N03/3705112800/ [Broken]

    I've tried using online sources like:

    I also looked up Binomial Theorem on wikipedia which just confused me more.
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Jul 9, 2009 #2


    User Avatar
    Homework Helper

    Well basically, the binomial theorem is valid for only n being an integer such that

    [tex](a+b)^n = b^n +^nC_1 b^{n-1}a + ^nC_2 b^{n-2}a^2 +...+ ^nC_r b^{n-r}a^r + a^n[/tex]

    But if you use the definition of nCr being n!/(n-r)!r!. Then terms like nC1 works out as

    [tex]^nC_1 = \frac{n!}{(n-1)!1!} = \frac{n(n-1)!}{(n-1)!}=n[/tex]

    [tex]^nC_2 = \frac{n!}{(n-2)!1!} = \frac{n(n-1)(n-2)!}{(n-1)!2!}= \frac{n(n-1)}{2}[/tex]

    So (a+b)n simplifies to this

    [tex](a+b)^b = b^n + nb^{n-1}a + \frac{n(n-1)}{2!}b^{n-2}a^2 + \frac{n(n-1)(n-2)}{3!}b^{n-3}a^3 + ...[/tex]

    so if ever you put n as a negative number or a fractional number then you'd have an infinite number of terms which is valid for |b/a| < 1.

    so in the chapter, the expansion is valid for

    [tex]\left | \frac{dx}{x} \right | < 1 \Rightarrow -1< \frac{dx}{x}<1[/tex]

    or simply dx/x is small, that is why they neglected the terms after x-3

    EDIT: Normally you won't need to do the y+dy = x +dx thing always, you can just usually apply y=xn => dy/dx=nxn-1
  4. Jul 9, 2009 #3


    User Avatar
    Science Advisor

    rockfreak667, if you go down far enough on the wikipedia page cited, you find "Newton's Generalized Binomial Formula" which does NOT require positive integers.

    For any non-negative integers, n and m, [itex]n\ge m[/itex]
    [tex]\left(\begin{array}{c}n \\ m\end{array}\right)= \frac{n!}{m!(n-m)!}= \frac{n(n-1)(n-2)\cdot\cdot\codt\(n-m+1)}{m(m-1)(m-2)\cdot\cdot\cdot(3)(2)(1)}[/tex]
    In that final form, we can, in fact, write it out for n and m any numbers, not just non-negative integers. Of course,
    [tex]\left(\begin{array}{c} n \\ n\end{array}\right)= 1[/itex]
    [tex]\left(\begin{array}{c}n \\ n-1\end{array}\right)= n[/itex]
    for any n, non-negative integer or not. Thus, we can write
    (x+ h)n= (1)xn+ (n)x{n-1}h+ terms involving h2 or higher, for any number n, positive integer or not. Then, (x+h)n- xn= n xn-1h+ terms involving h2 or higher and, dividing by h, [(x+h)n- xn]/h= n xn-1+ terms involving h or powers of h. Taking the limit as h goes to 0, all those "terms involving h or powers of h" go to 0 leaving only nxn-1.
  5. Jul 9, 2009 #4


    User Avatar
    Homework Helper

    That's how I learned the binomial theorem, first with n being a positive integer and then the generalized form for fractions and negative numbers. Most likely due to working out 50C34 and such we just used a calculator instead of writing it out...things like -3C1 on a calculator would make me go :bugeye:
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook