help with maclaurin/taylor series

[page89]

Hi guys,

I have recently been taught about the maclauren/taylor series and i have been set the following questions but i am struggling with them and i was hoping someone could help,

Q. Determine the Maclaurin series upto the term in x^3 for each of the following functions (you may assume Maclaurin series for standard functions)

part i) (1 + x) ^ (2/3)

part ii) x / (1 + 3x) ^ 3

part iii) (1 + x^2) sin x


I would be really gratefull if anyone can help me with them they have to be done for tomorrow

 

[galactus]

\(\displaystyle \frac{x}{(1+3x)^{3}}\)

\(\displaystyle \sum_{k=0}^{3}\frac{f^{k}(0)}{k!}=f(0)+f'(0)x+\frac{f''(0)}{2!}x^{2}+\frac{f'''(0)}{3!}x^{3}\)

\(\displaystyle f(0)=0\)

\(\displaystyle f'(0)=1\)

\(\displaystyle f''(0)=-18\)

\(\displaystyle f'''(0)=324\)

Assemble it:

\(\displaystyle x-9x^{2}+54x^{3}\)

 

[page89]

Thanks I knew it had something to do with differentiating each term but i think i went wrong trying fo differentiate it for the 3rd time.

Thanks.

Can anyone help with the others?

 

[Deleted member 4993]

Thanks I knew it had something to do with differentiating each term but i think i went wrong trying fo differentiate it for the 3rd time.

Thanks.

Can anyone help with the others?

Please show us your work - and exactly where you are stuck - so that we would know where to begin.