Question

I need help with the Taylor series. Finding first few non zero terms of a function. Something like ln(1-2x)

Answer 1

do you know the one for \(\ln(1+x)\)?

Answer 2

Yes something like 1-x+x^2-x^3

Answer 3

replace \(x\) by \(-2x\)

Answer 4

So should I just use something like substitution?

Answer 5

could you at least start me off in the right direction. I'm just having a hard time setting this up

Answer 6

we can do it by hand if you like, but i think it is easier to substitute

Answer 7

oh and btw your expansion of \(\ln(1+x)\) is not quite right
it should be
\[x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+...\] or
\[\sum_{n=1}^{\infty}\frac{(-1)^{n+1}x^n}{n}\]

Answer 8

then replace \(x\) by \(-2x\) and get
\[-2x-\frac{(2x)^2}{2}-\frac{(2x)^3}{3}-...\]