General Binomial Theorm

• July 27th 2011, 01:22 AM
Arron
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?
• July 27th 2011, 01:52 AM
FernandoRevilla
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 $t=-2x$ .
• July 27th 2011, 02:09 AM
Also sprach Zarathustra
Re: General Binomial Theorm
Quote:

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
• July 27th 2011, 03:37 PM
Arron
Re: General Binomial Theorm
What does that mean?
• July 27th 2011, 04:41 PM
mr fantastic
Re: General Binomial Theorm
Quote:

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.