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Not understanding textbook solution: Mary Boas mathematical methods

  1. Feb 17, 2014 #1
    1. The problem statement, all variables and given/known data

    4. Write the Maclaurin series for 1/√(1 + x) in ∑ form using the binomial coefficient
    notation. Then find a formula for the binomial coefficients in terms of n as we did
    in Example 2 above.


    2. Relevant equations

    [tex] { \left( 1+x \right) }^{ P }=\sum _{ n=0 }^{ \infty }{ \left( \underset { n }{ P } \right) } { x }^{ n } [/tex]

    3. The attempt at a solution

    This is what I got, [tex] \frac { 1 }{ \sqrt { (1+x) } } =\sum _{ n=0 }^{ \infty }{ \left( \underset { n }{ -1/2 } \right) } { x }^{ n } [/tex]

    This is the book's solution [tex] \left( \overset { -1/2 }{ n } \right) =\frac { { (-1) }^{ n }(2n-1)!! }{ (2n)!! } [/tex] I am not understanding the whole double factorial.
     
    Last edited: Feb 17, 2014
  2. jcsd
  3. Feb 17, 2014 #2
    They want you to write it out using the binomial coefficients. n!!=1*3*5*...*n where n is odd.
     
  4. Feb 17, 2014 #3

    LCKurtz

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    Use the \binom{}{} tex command for binomial coefficients:$$
    \binom{-\frac 1 2}{n}$$
     
  5. Feb 17, 2014 #4

    Ray Vickson

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    Write out the formula for ##{-1/2 \choose n}##. For example,
    [tex]{-1/2 \choose 0} = 1 \\
    {-1/2 \choose 1} = -1/2 \\
    {-1/2 \choose 2} = (-1/2)(-1/2 \:-1)/2! = \frac{1 \cdot 3}{2^2 \, 2!} [/tex]
    etc.

    The notation ##x!!## means ##x(x-2)(x-4) \cdots ##, ending at a final factor of 2 or 1 according as x is even or odd.
     
  6. Feb 17, 2014 #5
    Ah, now I see it [tex] (2k-1)!! = \prod_{i=1}^k (2i-1) [/tex]

    I have never seen that notation before. Thanks for the tip. Also, thanks for the LaTeX tip.

    Regards,
    Chris Maness
     
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