Why Does Using Power Series Help Approach the Classical Limit in Physics?

[CAF123]

Homework Statement

I have to show that the Planck radiation formula reduces to the Rayleigh-Jeans formula in the classical limit for blackbodies.

The Attempt at a Solution

I can easily show it using power series expansion of $e^{(hc/\lambda kT)}$ but I don't understand really why using a power series approximation makes something tend to the classical limit?

Similarly, for $E_k = mc^2(\gamma -1)$ tending to $\frac{mv^2}{2}$, in the classical limit. The results are clear, I just don't understand why using a power series actually works.

Many thanks.

 

[voko]

The first term (or terms) of a power series is a good approximation of a function only when its argument is small. Use any estimate of the approximation error to show that formally.

If you can cast "classicality" as a smallness of some argument to some function, then a power series (polynomial, actually) approximation would describe the phenomenon "classically". See how that applies to these two cases.

 

[CAF123]

The first term (or terms) of a power series is a good approximation of a function only when its argument is small. Use any estimate of the approximation error to show that formally.

What do you mean by the word 'argument' here?

 

[voko]

f(x) is function f of argument x.