Binomial expansion

Hi guys I've found the binomial expansion of a fraction which i attached here and the whole workings but at the end I'm not sure if the binomial expansion carries on or it should be stopped?i've marked both ways by a question mark so you can see the answers and help me choose one of them.
thanks for your time

Hi guys I've found the binomial expansion of a fraction which i attached here and the whole workings but at the end I'm not sure if the binomial expansion carries on or it should be stopped?i've marked both ways by a question mark so you can see the answers and help me choose one of them.
thanks for your time

It definitely carries on. Using the $+ \cdots \, \,$ is good practice. In A-Level exams, you'll be asked to "expand up to the term in x^2 or x^3 or whatever" and you can give your final answer without the ellipses.

Like Zacken says, it carries on forever but you are only asked to do up to a certain term. I always put a +... at the end, it doesn't matter if you don't but it shows that you understand that the expansion carries on.

how did you realise that it would carry on?the text book answer is opposite of yours .The question actually asked me to find the binomial expansion of the mentioned fraction on the attachment up to and including the term in x^2

i know for the negative power it carries on and for positive power it wouldn't carry on but here we have both of multiplied by together so that's why it confuses me

Well if you have something that doesn't carry on multiplied that does carry on, it's logical to think that the answer would be something that does carry on...

yeah i agree with you but here's the text book answer if you wanna take a look at it

The textbook isn't using $=$ signs. They're throwing away all the terms after the $+ \cdots$ and saying that the expansion is $\approx$ what they've written.

yeah that's what i thought as well but again at the end they use the = sign which i think is incorrect because the expansion carries on

No. They're saying the approximation is equal to that. I could write:

$1 \approx 0.5 + 0.4 = 0.9$, that means I'm saying that $1 \approx 0.9$ even if the last sign is an equals sign. It does not mean I'm saying $1 = 0.9$.

i know for the negative power it carries on and for positive power it wouldn't carry on but here we have both of multiplied by together so that's why it confuses me

too late again ...

If you are multiplying a finite series by an infinite one, yo get an infinite series. Think about it, you are multiplying an infinite number of terms by the finite series all the time, so it never ends.