# Math Help - General Binomial Theorm

1. ## General Binomial Theorm

I just need to verify my answer to the following problem

Use the General Bionomial Theorem to determine first four terms of the Taylor series at 0 for the function

f(x) = (1-2x)^-1/3

I have

By the General Bionomial Theorem

(1-2x)^-1/3

(-1/3)
( n ) x^n, for |x|<1

Where
(-1/3)
( n ) = (-1/3)(-4/3)(-7/3)....(-1/3-n=1) / n!

hence
(1-2x)^-1/3

= 1 + (-1/3)/1 -2x + (-1/3)(-4/3)/2! -2x^2 + (-1/3)(-4/3)(-7/3)/3! -2x^3 .....

= 1+ 2/3x - 4/9x^2 + 28/81x^3 ........

I also need to state the radius of convergence of this power series.

can anyone help?

2. ## Re: General Binomial Theorm

In general, the radius of convergence for the series $\sum_{n=0}^{+\infty}\binom{p}{n}t^n\;(p\in\mathbb{ R})$ is $1$ (why?). Now, substitute $t=-2x$ .

3. ## Re: General Binomial Theorm

Originally Posted by Arron
I just need to verify my answer to the following problem

Use the General Bionomial Theorem to determine first four terms of the Taylor series at 0 for the function

f(x) = (1-2x)^-1/3

I have

By the General Bionomial Theorem

(1-2x)^-1/3

(-1/3)
( n ) x^n, for |x|<1

Where
(-1/3)
( n ) = (-1/3)(-4/3)(-7/3)....(-1/3-n=1) / n!

hence
(1-2x)^-1/3

= 1 + (-1/3)/1 -2x + (-1/3)(-4/3)/2! -2x^2 + (-1/3)(-4/3)(-7/3)/3! -2x^3 .....

= 1+ 2/3x - 4/9x^2 + 28/81x^3 ........

I also need to state the radius of convergence of this power series.

can anyone help?
taylor &#40;1-2x&#41;&#94;-1&#47;3 at x&#61;0 - Wolfram|Alpha

4. ## Re: General Binomial Theorm

What does that mean?

5. ## Re: General Binomial Theorm

Originally Posted by Arron
What does that mean?
What does what mean? Please quote what you refer to in a thread.

If you're referring to post #3, then what it means is that you were shown what the answer is so that you can see whether your answer is correct (and if you're answer is not correct, then you can see what term is wrong and go back to your working and look for the mistake(s)).

If you're referring to post #2, then you need to be much more specific with what you don't understand.