TITLE: Find the limit of $(\sqrt2-\root3\of2 )(\sqrt2 -\root4\of2 )...(\sqrt 2-\root{n}\of2 )$ without using the squeezing principle
QUESTION [6 upvotes]: $$\mathop {\lim }\limits_{n \to \infty } (\sqrt 2  - \root 3 \of 2 )(\sqrt 2  - \root 4 \of 2 )...(\sqrt 2  - \root n \of 2 )$$
I was able easily to find the limit of this series using the squeezing principle, but how can you find it without it? I'm guessing it's an algebra trick :)

REPLY [2 votes]: Our sequence is $a_n = \prod_{k=1}^n (2^{1/2}-2^{1/k}).$
Take the log to get
$$\log(a_n) = \sum_{k=1}^n \log(2^{1/2}-2^{1/k}).$$
This is a typical thing to do when you have a product. Note that for $k$ large $2^{1/2}-2^{1/k}$ is near $\sqrt{2}-1$ and $\log(\sqrt{2}-1)<0$. Thus the series diverges and $\log(a_n) \to -\infty$. This means $a_n \to 0$.