Results 1 to 2 of 2

Math Help - a question about power series

  1. #1
    Junior Member
    Joined
    Nov 2008
    Posts
    54

    a question about power series

    integrate(arctan(x^2)) from 0--->1 with finding a numerical serie,what is the value which makes the error smaller than 0,0001??
    thanks for,ı need this as soon as possible.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Mathstud28's Avatar
    Joined
    Mar 2008
    From
    Pennsylvania
    Posts
    3,641
    Quote Originally Posted by sah_mat View Post
    integrate(arctan(x^2)) from 0--->1 with finding a numerical serie,what is the value which makes the error smaller than 0,0001??
    thanks for,ı need this as soon as possible.
    We know that

    \forall{x}\in[-1,1]~\arctan(x)=\sum_{n=0}^{\infty}\frac{(-1)^nx^{2n+1}}{2n+1}

    So

    \forall{x}\in[-1,1]~\arctan\left(x^2\right)=\sum_{n=0}^{\infty}\frac{  (-1)^nx^{4n+2}}{2n+1}

    So since (0,1)\subset[-1,1] and a power series is uniformly covnergent and integrable on the interior of its IOC we have that
    \begin{aligned}\int_0^1\arctan\left(x^2\right)dx&=  \int_0^1\sum_{n=0}^{\infty}\frac{(-1)^nx^{4n+2}}{2n+1}dx\\<br />
&=\sum_{n=0}^{\infty}\int_0^1\frac{(-1)^nx^{4n+2}}{2n+1}dx\\<br />
&=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)(4n+3)}<br />
\end{aligned}

    Now to find the appropriate error note that if S=\sum_{n=0}^{\infty}(-1)^a_n and S_n=\sum_{n=0}^{N}(-1)^na_n that

    R_N=\left|S-S_N\right|\leqslant{a_{N+1}}

    So the error of your series is less than or equal to the first neglected term, so calculate a_{N+1}\leqslant{.00001}
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Power Series Question
    Posted in the Calculus Forum
    Replies: 7
    Last Post: November 30th 2010, 08:55 PM
  2. [SOLVED] Power Series Question
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: July 15th 2010, 05:52 PM
  3. Replies: 0
    Last Post: January 26th 2010, 09:06 AM
  4. Help on a Power Series Question!
    Posted in the Calculus Forum
    Replies: 2
    Last Post: November 8th 2009, 04:48 PM
  5. Sequences and Series - Power Series Question
    Posted in the Calculus Forum
    Replies: 3
    Last Post: April 20th 2008, 08:32 PM

Search Tags


/mathhelpforum @mathhelpforum