# Maclaurin Series

1. Oct 29, 2005

### Bucky

obtain the maclaurin series expansions of the following:
$$ln(1+e^x)$$
ok im quite lost..i assume you set it equal to f(x) then differentiate..but what happens when you differentiate that?
also question (b) is
$$\frac{x}{(e^x-1)}$$
does that work down to
(e^x - 1)^-x ?

and if so..would you then differentiate the whole of the bracket? but then the chain rule comes into use doesnt it? so it would be -x times e^x?

Last edited: Oct 29, 2005
2. Oct 29, 2005

### finchie_88

I've had a go at the first part of your question, and I think I might have got somewhere with it but I'm not sure, this is what I did...
the maclaurin series for,
$$ln(1+x)=x-(0.5x^2)+(0.333x^3)-(0.25x^4)+...$$
$$ln(1+y) = y-(0.5y^2)+(0.333y^3)-(0.25y^4)+...$$
$$e^x = 1+x+((x^2)/2!)+((x^3)/3!)+...$$
y = e^x
when we combine them, we get something very complicated, but is something like...
Generalised result:
$$\ln(1+e^x) = kx+\frac{\(n^2x^2)(-1)^(n-1)}{2!}+\frac{\(n^3x^3)(-1)^(n-1)}{3!}+...$$

At the moment, I can't work out what k is, but I think I might have the rest of it (emphasis on the word think).
btw, Im crap at this maths symbol stuff, so if it comes out dodgy, make the most of it.

Last edited by a moderator: Oct 29, 2005
3. Oct 29, 2005

### Integral

Staff Emeritus
Do it and find out, follow the defintion of a Maclaurin's series. See what you get.
No,
$$\frac{x}{(e^x-1)}= x (e^x-1)^{-1}$$

use the quotient or product rule, should come out the same.

4. Oct 29, 2005

### Hurkyl

Staff Emeritus
You could say that f(x) = ln(1+e^x) and then try to find f'(x), f''(x), et cetera. That would be an equivalent problem.

How far have you gotten in trying to differentiate it? Do you remember your rules of differentiation?

5. Oct 30, 2005

### Bucky

ok heres how far i got:

f(x) = ln(1+e^x)

f'(x) = (1+e^x)^(-1)

f''(x) = -(1+e^x)^(-2)

f'''(x) = 2(1 + e^x)^(-3)

f^(iv)(x) = -6(1 + e^x)^(-4)

f^(v)(x) = 24(1+e^x)^(-5)

f(0) = ln(2)
f'(0) = 1/2
f''(0) = 4
f'''(0) = 16
f^(iv)(0) = 96
f^(v)(0) = 768

but now im not sure what to do. the notes get a bit hazy at this point

Last edited: Oct 30, 2005
6. Oct 30, 2005

### HallsofIvy

No, you haven't used the chain rule: now multiply by the derivative of 1+ ex which is ex:
$$f'(x)= \frac{e^x}{1+e^x}$$
Since your first derivative was wrong, this is wrong (and it's not even the derivative of (1+ex)-1: again you have not multiplied by the derivative of (1+ ex).
Use the quotient rule: the derivative of $\frac{e^x}{1+e^x}$ is
$$\frac{e^x(1+e^x)- e^{2x}}{(1+e^x)^2}= \frac{e^x}{(1+e^x)^2}$$
You should be able to see a pattern.
These are correct
These are incorrect

If your notes have the basic formula:
$$\Sigma_{n=0}^\infty \frac{f^{(n)}(0)}{n!}x^n$$
it should not be hazy at all.

Last edited by a moderator: Oct 30, 2005
7. Oct 30, 2005

### Bucky

ok im just rubbish at math it seems..
quotient rule is bottom times differentiated top minus top times differentiated bottom. all divided by bottom squared. right?
so
f'''(x) = (1+e^x)^2 e^x - e^x over
2(1+e^x) + e^x

f'''(x) = e^3x-e^x over
r2+3e^x

(latex ftw ¬.¬ )
i get the feeling im deeply wrong

i read on and i see how this works now, its just differentiating e to the power of stuff (which in practical terms i doubt ill need to know but ho hum..) and applying chain/quotient rules thats getting me.