# Doubt in definition of linear thermal expansion

The linear expansion model is an approximation. The more "complete" model assumes a small change $$\Delta L$$ in the length due to a small change in temperature from $$T$$ to $$T+\Delta T$$ is proportional to the product of $$\Delta T$$ with the length at temperature $$T$$, $$L(T)$$. If you choose the proportionality factor to be $$\alpha$$ and assume it doesn't depend on the temperature, in the limit of vanishingly small changes it's possible to show the length will depend on the temperature with an exponential behavior: $$L(T) = L_0 e^{\alpha T},$$ where $$L_0$$ is the length at $$T=0$$ on your temperature scale. If the argument in the exponential is small (and in nature we typically find values such that $$\alpha \ll 1$$), we may expand the exponential in it's series expansion and keep only the lowest terms: $$L=L_0 e^{\alpha T} = L_0 \left ( 1 + \alpha T + \frac{(\alpha T)^2}{2} + ... \right ) \approx L_0(1+\alpha T),$$ (if $$\alpha \ll 1$$, it's not hard to convince yourself that $$\alpha^2 \ll \alpha$$, and so forth for any higher power of $$\alpha$$, so it's legitimate to neglect those terms - provided $$T$$ is not too big). Since we have neglected all terms with power higher than one in $$\alpha$$, you should also neglect it on your calculations. So the two calculations are equal up to first order in $$\alpha$$, which is where the linear approximation works anyway.
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