1. Not finding help here? Sign up for a free 30min 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!

Binomial series expansion

  1. Jan 19, 2014 #1
    1. The problem statement, all variables and given/known data
    For ##n>0##, the expansion of ##(1+mx)^{-n}## in ascending powers of ##x## is ##1+8x+48x^{2}+...##

    (a) Find the constants ##m## and ##n##
    (b) Show that the coefficient of ##x^{400}## is in the form of ##a(4)^{k}##, where ##a## and ##k## are real constants.

    2. Relevant equations
    Binomial expansion

    3. The attempt at a solution
    I have expanded the series using binomial theorem, and compared each coefficient. Then I solved them using simultaneous equation. That works out to be ##m=-4, n=2##.

    (b) I know the binomial coefficient is ##\frac{n(n-1)(n-2)...(n-r+1)}{r!}##, though I'm stuck on that... :(
     
    Last edited: Jan 19, 2014
  2. jcsd
  3. Jan 19, 2014 #2

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    if ##n=2##, then how can there be an ##x^{400}## term? Why does the provided expansion not terminate?
    if m=-4, as well, then wouldn't that mean that the second term in the expansion is -8x instead of the +8x given?
     
    Last edited: Jan 19, 2014
  4. Jan 19, 2014 #3
    Oops... It should be ##(1+mx)^{-n}##... Sorry about that. :P
     
  5. Jan 19, 2014 #4

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    OK - so you have the expansion, in principle, of form:
    $$(1+mx)^{-n}=\sum_{i=1}^N a_ix^i$$ ... the coefficient of ##x^{400}## is ##a_{400}## .... and you know that ##a_i## depends on n,m and r from the binomial expansion formula.

    You have a formula for the binomial coefficient - but ##a_{400}## has an extra term due to the value of m.
    The trick is to match up the values you have with the formulas ... i.e. while n=2 in the equation above, the "n" in the formula for the binomial coefficient is different ... it may help to rewrite the formula taking this into account.
    If ##a_{400}## corresponds to the r'th binomial coefficient then what is r?
     
  6. Jan 19, 2014 #5
    I don't get it... :/
    I know the coefficient of ##x^{400}## would look something like this:
    ##\frac{-2(-3)(-4)....(-400-2+1)}{400!}##
    Though it doesn't look like what the question needs... :S
     
  7. Jan 20, 2014 #6

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    You forgot about the value of m ;)

    Anyway - you have to show that the series is the same as what they gave you.
    Find the value of a and k.

    Just to get you thinking the right way: is the product a positive or negative number?
    Maybe it is -2x3x4x...x401 or +2x3x4x...x401

    .... can you represent it in some kind of shorthand?
    Say using factorial notation?
     
  8. Jan 20, 2014 #7
    Where's the value of m? :confused:

    That actually looks like (401!) ? :confused:
     
  9. Jan 20, 2014 #8

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    You said that m=-4 remember?

    Notice that the question does not ask for the binomial coefficient.
    The coefficient of x^400 includes the binomial coefficient but is not equal to it... just like the coefficient of x^2 is not equal to it's binomial coefficient. Compare.

    That's right! So to ratio comes to 401!/400! doesn't it? ... what is that equal to?
     
  10. Jan 20, 2014 #9
    Still don't get it on the binomial coefficient though...

    I know that 401!/400!=(401x400!)/(400!)=401
     
  11. Jan 20, 2014 #10

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    That's all there is to it! Well done ;)
    How did you work out that it should be positive rather than negative?

    You need the part where "m=4" fits in to make the connection.
    Remember, you need to show that ##a_{400}=b(4)^k## where #a_n## is the coefficient of ##x^n## in the polynomial, with b and k both positive real numbers.
     
  12. Jan 20, 2014 #11
    I don't know about that though... :/

    But how do I use that m=4? :confused:
     
  13. Jan 21, 2014 #12

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    Well consider: what is the coefficient of x and what is the coefficient of x^2?
    How does that m=-4 contribute to those coefficients?
    How does the binomial coefficient contribute?

    You calculated it - you should know!
    Write ut the coefficient of x and x^2 in terms of n and m again - instead of actual numbers.
     
  14. Jan 21, 2014 #13
    Coefficient of x would look something like this: -nm
    Coefficient of x^2 would look something like this: ##\frac{(-n)(-n-1))}{2!} m^{2}##
     
  15. Jan 22, 2014 #14

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    Yes. m is just a multiplier on x, so you can start with the expansion for m=1 and just replace x with (mx) in the expansion.
     
  16. Jan 22, 2014 #15
    So that means... the coefficient of ##x^{400}## is
    ##\frac{(-2)(-3)(-4)...(-400-2+1)}{400!} (-4)^{400}## ?
     
  17. Jan 22, 2014 #16

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    Yes, but you can get rid of all of those awkward minus signs. Each extra factor like (-400-2+1) inverts the sign, but so does the extra -4 term, so they all cancel.
     
  18. Jan 22, 2014 #17
    Erm... why? How does it invert the minus sign? :confused:
    I know there are 400 terms, so the signs cancel each other, because 400 is an even number?
     
  19. Jan 22, 2014 #18

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    All the terms in the expansion are positive. (−2)(−3)(−4)...(−n−2+1) has n terms, so has the same sign as (-4)n.
     
  20. Jan 22, 2014 #19
    How does all terms in the expansion are positive? :confused:
     
  21. Jan 22, 2014 #20

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    The first term (1) is positive. To get the second term you multiply by (-2)(-4) = 8. To get the next you multiply be (-3)(-4) = 12. At every step you multiply by two factors, both negative, so the minus signs cancel.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Binomial series expansion
  1. Binomial expansions (Replies: 5)

  2. Binomial expansion (Replies: 1)

  3. Binomial Expansion (Replies: 6)

  4. Binomial expansion (Replies: 14)

  5. Binomial expansion (Replies: 2)

Loading...