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!

Finding a pattern for a series and the general formula

  1. Feb 11, 2013 #1
    1. The problem statement, all variables and given/known data
    Given:

    When n = 1:
    [itex]-1 + xln(e)[/itex]

    When n = 2:
    [itex]2 - 2xln(e) + x^{2}ln(e)^{2}[/itex]

    When n = 5:
    [itex]-120 + 120xln(e) - 60x^{2}ln(e)^{2} + 20x^{3}ln(e)^{3} - 5x^{4}ln(e)^{4} + x^{5}ln(e)^{5}[/itex]

    When n = 7:
    [itex]-5040 + 5040xln(e) - 2520x^{2}ln(e)^{2} + 840x^{3}ln(e)^{3} - 210x^{4}ln(e)^{4} + 42x^{5}ln(e)^{5} - 7x^{6}ln(e)^{6} + x^{7}ln(e)^{7}[/itex]

    Identify the series and write a general formula for it (in sigma summation notation if possible).


    2. Relevant equations
    N/A


    3. The attempt at a solution

    The pattern I do see is that the constant is always n!, but when n = 2, this factorial is not negative because there are an even number of terms, and the signs alternate starting from the highest power n where this term is positive.

    So the highest nth term does not have any factorials, but the subsequent terms, alternating signs, begin to increase in factorials based on what n is. For example, when n = 5, the next highest term has coefficient 5, the next term has 5*4, then 5*4*3, and finally 5*4*3*2.

    I'm not sure how to deal with the alternating signs in sigma summation notation or the factorials part.
     
  2. jcsd
  3. Feb 12, 2013 #2

    jbunniii

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    This might help with a formula for the coefficients:
    5*4*3*2*1 = (5!)/(0!)
    5*4*3*2 = (5!)/(1!) (which is also equal to (5!)/(0!))
    5*4*3 = (5!)/(2!)
    5*4 = (5!)/(3!)
    5 = (5!)/(4!)
    1 = (5!)/(5!)

    Also, (-1)^n = 1 if n is even, -1 if n is odd.

    So try writing something of the form
    $$\sum_{k = 0}^{n}(\textrm{formula involving factorials})(\textrm{formula involving }(-1)^k)(\textrm{formula involving power of }x \ln(e))$$
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Finding a pattern for a series and the general formula
  1. Find a general formula (Replies: 4)

Loading...