Find Taylor Series

1. Sep 27, 2013

TheFerruccio

This is rather embarrassing, because I should have known how to do this for years.

Question:

Compute the Taylor Series of $\frac{q}{\sqrt{1+x}}$ about x = 0.

Attempt at Solution:

Term-wise, I have gotten...

$f(0)+f'(0)+f''(0)+... = 1+1\left(-\frac{1}{2}\right)x+1\left(-\frac{1}{2}\right)\left(-\frac{3}{2}\right)x^2+1\left(-\frac{1}{2}\right)\left(-\frac{3}{2}\right)\left(-\frac{5}{2}\right)x^3+...$

I have gotten this to reduce to...

$\sum\limits_{k=0}^{\infty}x^k\left(-\frac{1}{2}\right)^k\frac{(2k-1)!!}{2^k}$

There is definitely a better way to do this. I am not thinking clearly. Additionally, I am not that confident in my answer, given the time of night.

2. Sep 27, 2013

Simon Bridge

At first glance - shouldn't there be a q in there?

If your termwise relation is supposed to be the series, you seem to have missed some bits.

Apart from that - it all seems pretty standard.

3. Sep 27, 2013

chingel

4. Sep 27, 2013

HallsofIvy

Staff Emeritus
Writing $q/\sqrt{1+ x}$ as $q(1+ x)^{-1/2}$ we have:
The value at 0 is q. The first derivative is $-(1/2)q(1+ x)^{-3/2}$ so
its value at 0 is -(1/2)q. The second derivtive is $(3/4)q(1+ x)^{-5/2}$ so
its value at 0 is (3/4)q. The third derivative is $-(15/8)q(1+ x)^{-7/2}$ so
its value at 0 is -(15/8)q. The fourth derivative is $(105/16)q(1+ x)^{-9/2}$ so
its value at 0 is (105/16)q...

That should be enough to see the pattern: it alternates sign with positive sign when n, the order of the derivative, is even so that is $(-1)^n$. The denominator of the fraction is a power of 2: $$2^n$$. There is, obviously, a factor of "q". The numerator of the fraction is the only "tricky" part. It is 1(3)(5)(7)..., a sort of "factorial" except that the even integers are missing. We can write that as 1(3)(5)(7)= 1(2)(3)(4)(5)(6)(7)/[2(4)(6)]= 7!/([2(1)][(2)(2)][(2)(3)])= 7!/[2^3(3!)] (when n= 4). So that is (2n- 1)!/[2^{n-1}(n-1)!].

Putting those together, the nth derivative, at x= 0, is
$$(-1)^n\frac{(2n-1)!}{2^{n-1}(2^n)(n- 1)!}q= (-1)^n\frac{(2n-1)!}{2^{2n-1}(n-1)!}q$$
and the coefficient of $x^n$ in the Taylor series expansion about x= 0 is that divided by n!.

Last edited by a moderator: Sep 27, 2013
5. Sep 27, 2013

mathman

You can get the same result, without calculating the derivatives, by getting the binomial expansion for (1+x)-1/2.

6. Sep 30, 2013

TheFerruccio

Sorry guys. I meant to type 1 instead of q. My keyboard broke and the function key messed up. I did not catch that error.

7. Sep 30, 2013

Simon Bridge

You have to know already that the binomial series is the Taylor series at x = 0 of the function f given by f(x) = (1 + x) α, where α ∈ ℂ is an arbitrary complex number.

I suppose that answers the original question.

8. Oct 1, 2013

mathman

The binomial theorem series and the Taylor series (for this case) are both power series around 0. Therefore they must be identical for the same function.

9. Oct 1, 2013

Office_Shredder

Staff Emeritus
If there was a simple explanation for why the binomial series worked for an arbitrary power then it would be a good answer, but as is it's basically just a way of telling someone what the answer is without explanation which isn't great.

10. Oct 2, 2013