You need to use the Maclaurin's series for the expansion of (1 + x)ⁿ

= 1 + nx + n(n - 1)/2! x² + ... + n(n - 1)...(n - r + 1)/r! xʳ + ...

But if we use it directly, the result will be in ascending powers of x.

So, factor out x first.

1/(x + 1)²

= (x + 1)⁻²

= (x(1 + x⁻¹))⁻²

= x⁻² (1 + x⁻¹)⁻²

Now the x term inside the brackets has a negative power.
So expanding it using Maclaurin's series will result in descending powers of x.

= x⁻² (1 + (-2)(x⁻¹) + (-2)(-3)/2! (x⁻¹)² + ...

= x⁻² (1 - 2x⁻¹ + 3x⁻² + ...)

= x⁻² - 2x⁻³ + 3x⁻⁴ + ...

Next,

The expression can be rewritten as :

1/x² - 2/x³ + 3/x⁴ + ...

Or 1/x² - 2(1/x³) + 3(1/x⁴)

①Notice that each term has a coefficient that is an integer, and this coefficient's absolute value is 1 less than the term's power/index of x


Eg. in 3/x⁴, 3 is less than 4 by 1.

Eg. in -2/x³, 2 is less than 3 by 1.


So for coefficient of 1/xʳ, the absolute value of the coefficient is (r - 1)


②Notice that the signs of each term alternate.

The first term has a positive sign
The second has a negative sign
The third term has a positive sign

And so on.

Odd-numbered terms have positive signs and even-numbered terms have negative signs.


③We notice that the term's parity (odd or even) is the opposite of its power's parity.

Odd terms have even powers, even terms have odd powers.

Which means even-power terms have positive signs and odd-power terms have negative signs.


Example :

1/x² is the first term (odd) but its power is even (the '2'). The sign is positive.

-2/x³ is the second term (even) but its power is odd (the '3'). The sign is negative

3/x⁴ is the third term (odd) but its power is even (the '4'). The sign is positive.


So,

What we can do is to introduce a factor of (-1)ʳ to represent/obtain the correct signs.


Even powers of -1 result in positive signs while odd powers of -1 result in negative signs.

Eg.

For 1/x², (-1)² = (-1)(-1) = 1
This makes the coefficient positive.

For -2/x³, (-1)³ = (-1)(-1)(-1) = -1
This makes the coefficient negative.

For 3/x⁴ , (-1)⁴ = (-1)(-1)(-1)(-1) = 1
This makes the coefficient positive.

Therefore, the coefficient of 1/xʳ is :

(r - 1)(-1)ʳ

Wait shouldn’t it be (-1)^( r-2)(r-1) since r more than or equals to 2

The r ≥ 2 just means that the term with highest power of x is 1/x².

Also realise that subtracting a 2 will not change the parity of the power and the sign is not affected.


Eg.


If you used (r - 1)(-1)ʳ⁻² for the coefficient.


When r = 3 ,

The term is (3 - 1)(-1)³⁻² x⁻³

= 2(-1)¹ /x³

= -2/x³

If you had used (r - 1)(-1)ʳ, it would be


2(-1)³ / x³

= -2/x³


It's the same. So there is no need for -2 when it's asking for simplest form.

To add on,

(-1)ʳ⁻² = (-1)ʳ/(-1)² = (-1)ʳ/1 = (-1)ʳ

Thanks I get it now :D

I found another method could u help me see if it will be accepted
I’ll post it now