Wikipedia Raabe's test proof doubt on second part

Question

I am trying to understand the second part of the following proof of Raabe's test.

Question:

1) How does the author finds out that $(1+\frac{R}{N})...(1+\frac{R}{n})\geqslant cn^R$ ? I understand the derivation before thanks to another thread but I do not see the relation to the $cn^R$. Where does that come from?

Thanks in advance!

Answer 1

Since $$\log\left(\left(1+\frac RN\right)\cdots\left(1+\frac Rn\right)\right)=R\log n+\mathcal O(1),$$ taking exponentials gives $$\left(1+\frac RN\right)\cdots\left(1+\frac Rn\right)=e^{R\log n+\mathcal O(1)}=e^{\mathcal O(1)}\cdot n^R$$ and $\exp(\mathcal O(1))$ is always greater than some constant $c$.