I've recently been reading about asset returns properties and I found in a book an statement that says that $[\prod_{j=0}^{k-1}(1+R_{t-j})]^\frac{1}{k}-1$ can be approximated as $(\frac{1}{k})\sum_{j=0}^{k-1}R_{t-j}$ using a first-order Taylor expansion.
I've tried to reach that result using the Taylor expansion definition and the derivative of a product, but I don't get even close to it. Could someone give me a hint to continue the proof, please?
Consider$$P=\Big[\prod_{j=0}^{k-1}(1+R_{t-j})\Big]^\frac{1}{k}\implies \log(P)=\frac{1}{k}\log\Big[\prod_{j=0}^{k-1}(1+R_{t-j})\Big]=\frac{1}{k}\sum_{j=0}^{k-1}\log(1+R_{t-j})$$ Now, by Taylor, assuming $R_{t-j} \ll 1$, $\log(1+R_{t-j})\sim R_{t-j}$. So $$\log(P)\sim \frac{1}{k}\sum_{j=0}^{k-1} R_{t-j}$$ and continue with Taylor $$P=e^{\log(P)}\sim 1+\frac{1}{k}\sum_{j=0}^{k-1} R_{t-j}$$ and you look for $(P-1)$