Rate of convergent of real Sequence.

Question

Consider the following two sequences of reals $$a_n=\frac{1}{n+1}+\cdots +\frac{1}{2n}$$ and $$b_n=\frac{1}{n}$$

Which of the following statement is true?

$1$. $\{a_n\}$ convergent to $log(2)$ and has the same convergent rate as $\{b_n\}$.

$2$. $\{a_n\}$ convergent to $log(4)$ and has the same convergent rate as $\{b_n\}$.

$3$. $\{a_n\}$ convergent to $log(2)$ but does not have the same convergent rate as $\{b_n\}$.

$4.$ $\{a_n\}$ does not converge.

I only know that $a_n$ is convergent to $log(2)$ and $b_n$ to $0$. How about rate of convergent?

Answer 1

To compare the rates of convergence of $\{a_n\}$ and $\{b_n\}$, one should compare how fast the sequences $\{a_n - \log 2\}$ and $\{b_n\}$ approach $0$. One can check that $$\lim_{n\to\infty} \frac{a_n - \log 2}{b_n} = \lim_{n\to\infty} \sum_{r=1}^n \frac{n}{n+r} - n\log 2 = -\frac{1}{4} \ne 0, \pm\infty$$ i.e., $\{a_n\}$ and $\{b_n\}$ share the same rate of convergence (if their rates of convergence are different, then $\lim_{n\to\infty} \frac{a_n - \log 2}{b_n}$ is $0$ or $\pm\infty$.)