# Taylor expansion

#### robinegberts

Hello friends,

I need to compute the taylor expansion of

$$\frac{x^4 e^x}{(e^x-1)^2},$$
for $x<<1$, to find

$$x^2 + \frac{x^4}{12}.$$

Can someone explain this to me?

Thanks!

Related Calculus and Beyond Homework News on Phys.org

#### mfb

Mentor
What does your textbook say about Taylor expansions? Did you follow its steps, where did you run into problems?

I don't know if it is actual homework, but I think it fits better to our homework section.

#### robinegberts

Yes, thank you mfb.

I tried:

$$x^4 \frac{1+x+O(x^2)}{(x+O(x^2))^2} = x^2 + x^3,$$
which is clearly not the way to go. My main question is how to deal with that the expansion of $e^x-1$ will be squared and how that influences my choice of developing up till certain order terms. Can this even be done using standard series?

#### mfb

Mentor
You'll need more terms in the expansion of the denominator.

Or calculate the derivatives, that takes more time but is less error-prone.

#### pixel

One thing I noticed is that if we write out the denominator we get

(ex-1)2 = e2x - 2ex + 1.

If we expand each exponential term to just first order in x, we get: 1 + 2x - 2 - 2x + 1 which = 0, whereas doing it as above you got x2.

#### Ray Vickson

Homework Helper
Dearly Missed
Yes, thank you mfb.

I tried:

$$x^4 \frac{1+x+O(x^2)}{(x+O(x^2))^2} = x^2 + x^3,$$
which is clearly not the way to go. My main question is how to deal with that the expansion of $e^x-1$ will be squared and how that influences my choice of developing up till certain order terms. Can this even be done using standard series?
That IS the way to go, but first you need to use the correct expansions for $e^x$, and you should keep 3 or 4 terms in the numerator and the enominator, just to be absolutely sure you have not neglected something important. Then, of course, you expand the denominator as
$$(1+Y)^{-2} = 1 - 2Y + [(-2)(-3)/2!]Y^2 +[(-2)(-3)(-4)/3!] Y^3 + \cdots,$$
where $1+Y$ is what you have left after expanding $e^x - 1$ and taking out the factor $x$.

#### mfb

Mentor
One thing I noticed is that if we write out the denominator we get

(ex-1)2 = e2x - 2ex + 1.

If we expand each exponential term to just first order in x, we get: 1 + 2x - 2 - 2x + 1 which = 0, whereas doing it as above you got x2.
That is fine, as the expansion does indeed start with 0 + 0x, you just missed the first nonzero term in your expansion.

To get three terms in the final expansion, we need three nonzero terms in the denominator, which means everything up to x4.

#### robinegberts

Clever Ray! Let's see whether I am getting it.

$$x^4 \frac{1+x+x^2/2}{(x+x^2/2+x^3/6)^2} = x^2 (1+x+x^2/2) (1+x/2+x^2/6)^{-2} = x^2 (1+x+x^2/2) (1-2(x/2+x^2/6)) = x^2 (1+x+x^2/2) (1-x-x^2/3) = (x^2+x^3+x^4/2)(1-x-x^2/3)=x^2-x^3-x^4/3+x^3-x^4+x^4/2=x^2-10/12 x^4.$$

Nearly there, did I miss something?

#### mfb

Mentor
You'll need one more order for the (1+c)-2 expansion: it adds x2 terms, which then lead to x4 terms in the final result.

WolframAlpha gets -1/12 as prefactor for x4.

#### ehild

Homework Helper
$$f(x) = \frac{x^4 e^x}{(e^x-1)^2}$$ can be written in a form easier to expand.
Divide numerator and denominator with ex.
$$f(x) = \frac{x^4 }{(e^{x/2}-e^{-x/2})^2}=\frac{ x^4 }{4 \sinh^2(x/2)}$$
Using that $\sinh^2(x/2)=\frac{\cosh(x)-1}{2}$, $$f(x)=\frac{x^4}{2(\cosh(x)-1)}$$.
It is very easy to expand the cosh(x)-1 function. cosh(x)'=sinh(x), and sinh(x)' = cosh(x). Expand to the fourth power in x. It is two terms only as cosh is an even function.

#### robinegberts

Thanks for the help mfb!I give up... I am studying for a thermal physics test, and this really does not contribute to my understanding anymore. ;-)

#### robinegberts

Ok, so that yields me

$$f(x) = \frac{x^4}{x^2+x^4/12},$$ now what? ;-)

#### mfb

Mentor
Just one step away:$$f(x) = \frac{x^4}{x^2+x^4/12} = x^2(1+x^2/12)^{-1} \approx x^2(1-x^2/12)$$

#### robinegberts

Ah, I understand. Wish I had more practise with these expansion. Thanks for the help all!

#### SammyS

Staff Emeritus
Homework Helper
Gold Member
One thing I noticed is that if we write out the denominator we get

(ex-1)2 = e2x - 2ex + 1.

.
Start with $\displaystyle \ \frac{x^4 e^x}{(e^x-1)^2}=\frac{x^4 }{e^{-x}(e^x-1)^2} \ .$

Then using pixel's expansion we have $\displaystyle\frac{x^4 }{e^{-x}(e^{2x}-2e^x+1)} \ .$

.And multiplying through by e−x we have $\displaystyle\frac{x^4 }{(e^{x}-2+e^{-x})} \ ,$

which easily leads to ehild's result if that's where you would like to go.

However, using the well known series for ex gives the following quite directly.

$\displaystyle\frac{x^4 }{\displaystyle 2\left(\frac{x^2}{2} + \frac{x^4}{4!} + \frac{x^6}{6!}+\dots \right)} \ ,$​
.
Use as many terms as necessary.

"Taylor expansion"

### Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving