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

In general:

(a+b)n=an+nan-1b+n(n-1)an-2b2/2!+n(n-1)(n-2)an-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!+...+nCkrk+...], where nCk (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.

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