Maclaurin expand

Expanding ln(1+2x)^3 / (1+x) up to x^3 term

I got 8x^3 / 1+x is this correct?

THanks!

I do not think so

for $\displaystyle f(x)=\frac{ln (1+2x)^3}{1+x}$

I got $6x-15x^2+26x^3$

and for $\displaystyle f(x)=ln \frac{(1+2x)^3}{1+x}$

I got $2x-\frac{11}{2}x^2+\frac{23}{3}x^3$

Expanding ln(1+2x)^3 / (1+x) up to x^3 term

I got 8x^3 / 1+x is this correct?

THanks!

You're looking for a series of the form

$A + Bx + Cx^2 + Dx^3$

- you shouldn't have any divisions left over at the end!

Expanding ln(1+2x)^3 / (1+x) up to x^3 term

I got 8x^3 / 1+x is this correct?

THanks!

I do not think so

for $\displaystyle f(x)=\frac{ln (1+2x)^3}{1+x}$

I got $6x-15x^2+26x^3$

and for $\displaystyle f(x)=ln \frac{(1+2x)^3}{1+x}$

I got $2x-\frac{11}{2}x^2+\frac{23}{3}x^3$

I only did the 1st series, but got 6x-12x^2+20x^3

You're looking for a series of the form

$A + Bx + Cx^2 + Dx^3$

- you shouldn't have any divisions left over at the end!

now, that i think about it, you just kind of simplified your expression in the limit as x goes to infinity, but even then it would rather be 8x^2, not what you said, you only simplified the top of the expression

For this, i must use the formula to expand?
Is there not a shortcut i could use?
Like substitution of 2x?

..

With questions like this for A-level, is it expected to solve them as a Maclaurin expansion (i.e. find f(0), f'(0), f''(0), etc) or is "any method that gets a valid power series" OK?

[Here I would use standard expansions and multiply out[.

For this, i must use the formula to expand?
Is there not a shortcut i could use?
Like substitution of 2x?

When you say formula I guess you mean differentiating the function a few times and sub in 0, that how you get the coefficients in front of the different powers of x in the series. Thats how Ive done mine anyway.
How would you use the 2x substitution, I've never heard of that method.
Maybe just tell us how you arrived at your solution and then we can explain what went wrong.

It might help for a start if you clarified what your function actually is!

Is it

$\displaystyle ln[\dfrac{(1+2x)^3}{1+x}]$

or

With questions like this for A-level, is it expected to solve them as a Maclaurin expansion (i.e. find f(0), f'(0), f''(0), etc) or is "any method that gets a valid power series" OK?

[Here I would use standard expansions and multiply out[.

Could i just ask, how wold you use standard expressions here?

Thanks!

You know the standard power series for ln(1 + A) and 1/(1 + y) for appropriate values of A and y so you can just multiply the power series and collect the terms you need BUT check on the ranges of convergence of each series to get the overall range of convergence.