# Proving Renyi entropy properties

I want to prove that Shannon entropy is a special case of Renyi entropy by solving this, $$\lim_{\alpha\to1}\frac{1}{1-\alpha}\ln\sum_{k=1}^{n}p_{k}^{\alpha} = -\sum_{k}p_{k}\ln p_{k}$$

Where $(a)$ follows from noting the limit $(1)$ is in the $\frac{0}{0}$ form as $\sum p_i = 1$. The rest is simple when you recall that $\frac{\mathrm{d}}{\mathrm{d}x} a^x = a^x \ln a.$
You should note, btw, that most of the time both Renyi and Shannon entropies are defined using $\log_2$ instead of $\ln$. This doesn't really alter anything, as both sides can be converted to the appropriate base by multiplying with a constant, and the above goes through with no change.