Results 1 to 6 of 6

Math Help - Constructing a Power Series using a geometric series as a base.

  1. #1
    Member
    Joined
    Oct 2009
    From
    Detroit
    Posts
    195
    Thanks
    5

    Constructing a Power Series using a geometric series as a base.

    i am to use \frac{1}{1-x}=\sum{x^n}

    to find a power series representation for \frac{1+x}{1-x}

    additionally I need to find the interval of convergence.

    since, \frac{1}{1-x}=\sum{x^n}

    and

    (1+x)\frac{1}{1-x}=\frac{1+x}{1-x}

    then

    (1+x)*\sum{x^n}


    the book says 1+2\sum{x^n} is a power series representation of the original function \frac{1+x}{1-x} can someone point out my error?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,561
    Thanks
    785

    Re: Constructing a Power Series using a geometric series as a base.

    (1+x)*\sum{x^n} is correct, but it is not in the form \sum c_nx^n.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Oct 2009
    From
    Detroit
    Posts
    195
    Thanks
    5

    Re: Constructing a Power Series using a geometric series as a base.

    we'll i don't think i can use the distributive property to make it
    \sum{x^n}+\sum{x^{n+1}}
    since this is basically an polynomial of infinitely many terms being multiplied by a binomial.

    i don't really see anything to do here.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Oct 2009
    From
    Detroit
    Posts
    195
    Thanks
    5

    Re: Constructing a Power Series using a geometric series as a base.

    i don't know of a way to dump out the parentheses and make a " c_n"
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,561
    Thanks
    785

    Re: Constructing a Power Series using a geometric series as a base.

    Quote Originally Posted by bkbowser View Post
    we'll i don't think i can use the distributive property to make it
    \sum{x^n}+\sum{x^{n+1}}
    since this is basically an polynomial of infinitely many terms being multiplied by a binomial.
    This depends on the level of rigor with which you need to "find" the series for \frac{1+x}{1-x}. Do you need an accurate proof that it converges to the function inside the interval of convergence? I agree that infinite sums require special care, but using distributivity in the way you did is fine as the first approximation.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    12,116
    Thanks
    1000

    Re: Constructing a Power Series using a geometric series as a base.

    \frac{1+x}{1-x}

    \frac{1}{1-x} + \frac{x}{1-x}

    (1 + x + x^2 + x^3 + ... ) + (x + x^2 + x^3 + x^4 + ...)

    1 + 2x + 2x^2 + 2x^3 + ...

    1 + 2(x + x^2 + x^3 + ... ) = 1 + 2\sum_{n=1}^\infty x^n
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: May 22nd 2012, 06:57 AM
  2. Replies: 3
    Last Post: September 29th 2010, 07:11 AM
  3. Replies: 0
    Last Post: January 26th 2010, 09:06 AM
  4. geometric power series
    Posted in the Calculus Forum
    Replies: 4
    Last Post: December 8th 2009, 10:30 AM
  5. Replies: 10
    Last Post: April 18th 2008, 11:35 PM

Search Tags


/mathhelpforum @mathhelpforum