Results 1 to 5 of 5

Thread: [SOLVED] Maclaurin Series

  1. #1
    Member billa's Avatar
    Joined
    Oct 2008
    Posts
    100

    [SOLVED] Maclaurin Series

    I was trying to do this problem in my book

    find the Maclaurin Series for atan(x^2) using the table found on the previous page

    The table listed the series for atan(x) and they apparently wanted you to plug x^2 in for x to use the table to create the series. I am trying to figure out why you can make this substitution - I know that it works but the reason why didn't jump out at me. So I tried to use the definition of Maclaurin series,

    <br />
\sum\limits_{n = 0}^\infty  {\frac{{f^{(n)} (0)}}<br />
{{n!}}} x^n  = f(x)<br />


    to figure out why it was ok, but I couldn't. So my question is, why can you substitute another function of x into the definition of a Maclaurin and/or Taylor Series?

    Here is what I tried to do - I let u(x) be x^2 and tried to show the following

    <br />
\sum\limits_{n = 0}^\infty  {\frac{{f^{(n)} (u(0))}}<br />
{{n!}}} x^n  = \sum\limits_{n = 0}^\infty  {\frac{{f^{(n)} (0)}}<br />
{{n!}}} u^n <br />


    but I don't know enough about these series to work that out
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    14,784
    Thanks
    3006
    how about this version ?

    \frac{1}{1-x} = 1 + x + x^2 + x^3 + ...

    \frac{1}{1 + x^2} = \frac{1}{1 - (-x^2)} = 1 - x^2 + x^4 - x^6 + ...

    integrate ...

    \arctan(x) = C + x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + ...

    since \arctan(0) = 0 ... C = 0

    so ...

    \arctan(x^2) =  x^2 - \frac{x^6}{3} + \frac{x^{10}}{5} - \frac{x^{14}}{7} + ...
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member billa's Avatar
    Joined
    Oct 2008
    Posts
    100
    Ty, but my question was why can you do the last step - I solved the problem I just don't know why you can take atan(x) and put in x^2 for x.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    14,784
    Thanks
    3006
    Quote Originally Posted by billa View Post
    Ty, but my question was why can you do the last step - I solved the problem I just don't know why you can take atan(x) and put in x^2 for x.
    because it's a composite function.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member billa's Avatar
    Joined
    Oct 2008
    Posts
    100
    Wow, what was going on with my brain? I don't even know what I was trying to do there - good thing I didn't ask my teacher lol
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Maclaurin Series.?!
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Mar 18th 2010, 09:23 AM
  2. Replies: 0
    Last Post: Jan 26th 2010, 08:06 AM
  3. [SOLVED] Maclaurin Series
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Aug 22nd 2009, 04:32 AM
  4. Replies: 4
    Last Post: Nov 17th 2008, 11:53 PM
  5. [SOLVED] Maclaurin series for square root
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Feb 2nd 2008, 01:07 PM

Search Tags


/mathhelpforum @mathhelpforum