# Not understanding textbook solution: Mary Boas mathematical methods

1. Feb 17, 2014

### kq6up

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

$${ \left( 1+x \right) }^{ P }=\sum _{ n=0 }^{ \infty }{ \left( \underset { n }{ P } \right) } { x }^{ n }$$

3. The attempt at a solution

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

This is the book's solution $$\left( \overset { -1/2 }{ n } \right) =\frac { { (-1) }^{ n }(2n-1)!! }{ (2n)!! }$$ I am not understanding the whole double factorial.

Last edited: Feb 17, 2014
2. Feb 17, 2014

### crownedbishop

They want you to write it out using the binomial coefficients. n!!=1*3*5*...*n where n is odd.

3. Feb 17, 2014

### LCKurtz

Use the \binom{}{} tex command for binomial coefficients:$$\binom{-\frac 1 2}{n}$$

4. Feb 17, 2014

### Ray Vickson

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

5. Feb 17, 2014

### kq6up

Ah, now I see it $$(2k-1)!! = \prod_{i=1}^k (2i-1)$$

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

Regards,
Chris Maness