# 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.