C4 Binomial expansion - values for which expansions are valid?

My textbook goes on about finding the values for which the expansion is valid, usually mod x < 1, but I have no idea how to get it or what it means... Could anyone explain what it means and how do I find it out for a particular expansion?

For example, (1-x)^1/3

How would you know what values it is valid for?

(edited 10 years ago)

Reply 1

joostan

13

Original post by lollage123

My textbook goes on about finding the values for which the expansion is valid, usually mod x < 1, but I have no idea how to get it or what it means... Could anyone explain what it means and how do I find it out for a particular expansion?

For example, (1-x)^1/3

How would you know what values it is valid for?

Take:

\ (1+ax)^n

Here the

ax

is the "

x

" for which

|x| < 1

So:

|ax| < 1 \Rightarrow x<...

?

Reply 2

Smaug123

15

Original post by lollage123

My textbook goes on about finding the values for which the expansion is valid, usually mod x < 1, but I have no idea how to get it or what it means... Could anyone explain what it means and how do I find it out for a particular expansion?

For example, (1-x)^1/3

How would you know what values it is valid for?

The reason is that the series for (1+x)^n goes like:

1 + nx+\frac{n(n-1)}2x^2+…

If mod x >= 1, it's clear that this will never converge because x^2 > x, x^3 > x^2, …
But if mod x < 1, it has a chance of converging because x^2 < x, etc - the terms are getting smaller. It can be shown rigorously that indeed the series converges for mod x < 1 (but that's a big kettle of fish involving analysis and the ratio test).

C4 Binomial expansion - values for which expansions are valid?

Quick Reply

Related discussions

Latest

Trending

Trending

Articles for you