Limit and sum question

Question

Solve these limits.

$$\lim_{n\to\infty}\left(\,\frac{1}{n}+\frac{1}{n+1}+\ldots+\frac{1}{2n}\,\right)=?$$

and $\lim_{n \to \infty}a_{n} = ?$ where

$$a_{0}=1\,,\quad a_{n}=\frac{1}{2}\left(\,\frac{2}{a_{n-1}}+a_{n-1}\,\right)$$

Answer 1

hiint: For the first one you can use the Riemann sum

$$ \frac{1}{n}\sum_{k} \frac{1}{1+k/n} \longrightarrow_{n\to \infty} \int_{0}^{1}\dots dx$$

For the second one, assume $\lim a_n = b$ and subs back in the eq. and solve for $b$.

Answer 2

Problem 1: compare $$ \lim_{n \to \infty} \sum_{k=n}^{2n}\frac{1}{k} $$ to $$ \lim_{n \to \infty} \int_{n}^{2n}\frac{dx}{x}=\log 2 $$

Problem 2: if the limit exists, it is equal to $$ x=\frac{1}{x} + \frac{x}{2} $$ Can you handle from here?