4.1.4 Binomial Approximation

I have seen a few questions that uses the well known first-order binomial approximation of:

(1+x)n≈1+nx, if ∣xn∣≪1(1 + x)^n \approx 1 + nx, \text{ if } |xn| \ll 1

These will typically be approximation questions (as the identity itself is an approximation) and, because xn≪1xn \ll 1, the questions usually has this value multiplied by a large integer in order to give a sufficient range of answers. An example question would be:

1000(1.0002)50≈1000[1+(.0002×50)]=1000(1.01)=10101000(1.0002)^{50} \approx 1000[1 + (.0002 \times 50)] = 1000(1.01) = 1010

This answer is incredibly close to the exact answer of 1010.049…1010.049 \dots. A natural question that arises is how much does ∣xn∣|xn| need to be less than 1 in order to use it? There is no easy answer to this, but I’d figure that if the test writers have a problem that looks like you’d be able to use the approximation, then you are probably OK to use it!