335 views

138 views

### Find the limit of the following sequence. (Real Analysis) [duplicate]

Find the limit of the following sequence. $$a_n = \frac{\sqrt[n]{(n+1)(n+2)...(2n)}}{n}$$ I tried couple of methods: Stolz, Squeeze, D'Alambert. But I can not seem to make a conclusion on the limit.
83 views

### To evaluate limit of sequence $\left(\left( 1 + \frac1n \right) \left( 1 + \frac2n \right)\cdots\left( 1 + \frac nn \right) \right)^{1/n}$
How do I evaluate the limit of the following sequence $$a_n = \left(\left( 1 + \frac1n \right) \left( 1 + \frac2n \right)\cdots\left( 1 + \frac nn \right) \right)^{1/n}$$ I tried to take log and ...
### Limit of ${a_n}^{1/n}$ is equal to $\lim_{n\to\infty} a_{n+1}/a_n$
Today my lecturer put up on the board that: If $\lim\limits_{n\to\infty}\frac{a_{n+1}}{a_n}$ exists and $a_n>0$ then \$\displaystyle \limsup\limits_{n\to\infty}\left(a_n^{\frac{1}{n}}\right)=\...