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!

Find the expansion of this term

  1. Sep 18, 2014 #1

    utkarshakash

    User Avatar
    Gold Member

    1. The problem statement, all variables and given/known data
    IF [itex]e^{m \arctan x}=a_0 + a_1x + a_2x^2 + a_3x^3.......+a_nx^n+..........[/itex]
    prove that [itex](n+1)a_{n+1} + (n-1)a_{n-1}=ma_n [/itex]
    and hence obtain the expansion of [itex]e^{m \arctan x} [/itex].

    2. Relevant equations

    3. The attempt at a solution
    $$e^{m \arctan x} = 1+m \arctan x + (m \arctan x)^2/2! + (m \arctan x)^3/3! + ..........$$
    But I need to simplify this expansion in terms of x and I'm clueless how to do that.
     
  2. jcsd
  3. Sep 18, 2014 #2

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    How did you get the terms of that series?
     
  4. Sep 18, 2014 #3

    Mark44

    Staff: Mentor

    The Maclaurin series expansion of emy is
    $$1 + my + \frac{(my)^2}{2!} + \dots + \frac{(my)^n}{n!} + \dots$$

    Replace y with arctan(x) to get the expansion that utkarshakash shows.

    I've been looking at this problem, but no joy as yet.
     
  5. Sep 18, 2014 #4

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    You cannot just substitute any arbitrary function for y ... consider: what if y=ln|x|/m ?
    You have to work out the series coefficients by actually doing the differentiation. It's not as hard as it's, at first, seeming.
     
  6. Sep 18, 2014 #5

    utkarshakash

    User Avatar
    Gold Member

    Got it! Thanks a lot!
     
  7. Sep 18, 2014 #6

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    No worries - if a shortcut does not get you where you need to be, try going the long way round ;)
     
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: Find the expansion of this term
  1. Finding 2012th term (Replies: 21)

Loading...