(X+h)^-2 expand

Question

Expand the following

Answer 1

(x+h)^-2 can be expanded using the binomial theorem into an infinite series.

In general:

(a+b)ⁿ=aⁿ+na^n-1b+n(n-1)a^n-2b²/2!+n(n-1)(n-2)a^n-3b³/3!+...

This can also be written:

aⁿ(1+(b/a))ⁿ, then let r=b/a (for convenience), hence the expansion:

aⁿ[1+nr+n(n-1)r²/2!+n(n-1)(n-2)r³/3!+...+ⁿC_kr^k+...], where ⁿC_k (combination function)=n!/((n-k)!k!) and 0≤k≤n. Note that the combination function only makes sense when n>0.

If a=x, r=h/x and n=-2, by substitution we get:

x^-2(1+(h/x))^-2=(1/x²)(1-2(h/x)+(-2)(-3)(h/x)²/2!+(-2)(-3)(-4)(h/x)³/3!+(-2)(-3)(-4)(-5)(h/x)⁴/4!+...), an infinite series because the factors in the arithmetic multiplier just keep increasing negatively.

Note that, if h is small compared to x, a good approximation of (x+h)^-2 is (1/x²)(1-2h/x)=1/x²-2h/x³. This approximation is useful in determining the derivative of x^-2, which is -2/x³ or -2x^-3.