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## Limits of Sequences and Series

Infinitely many mathematicians walk into a bar. The first says "I'll have a beer". The next ones say "I'll have half of the previous guy". The bartender pours out 2 beers and says "Know your limits".

# Level 4

$\large \displaystyle \lim_{n \to \infty} \frac { \tan \left (89. \underbrace{9999999 \ldots 9^\circ}_{n \text{ 9's}} \right )} {\tan \left (89. \underbrace{9999999 \ldots 9^\circ}_{(n-1) \text{ 9's}} \right )} = \ ?$

A sequence adheres to the recursion relation $$a_{n}=\sqrt[3]{a_{n-1}a_{n-2}a_{n-3}}$$ with initial terms $$a_{0}=1,a_{1}=2,a_{2}=1$$.

As $$n$$ tends to infinity, what is the limit of $$a_n$$ correct to 2 decimal places?

Evaluate

$\sqrt{\dfrac{1}{2}}\times\sqrt{\dfrac{1}{2} + \dfrac{1}{2}\sqrt{\dfrac{1}{2}}}\times\sqrt{\dfrac{1}{2} + \dfrac{1}{2}\sqrt{\dfrac{1}{2} + \dfrac{1}{2}\sqrt{\dfrac{1}{2}}}}\cdots$

The sequence $$\{ {a}_{n}\}$$ satisfies $$\left(\frac{2n+1}{2n-1}\right)^{a_n} =e^2.$$ Find $$\lim_{n \rightarrow \infty}\frac{2a_n}{n+1}$$.

The sequence $$1, 2, 3, 2, \ldots$$ has the property that every fourth term is the average of the previous three terms. That is, the sequence $$\{a_n\}$$ is defined as $$a_1 = 1, a_2 = 2, a_3 = 3$$, and $a_{n+3} = \dfrac{a_{n+2} + a_{n+1} + a_n}{3}$ for any positive integer $$n$$.

This sequence converges to a rational number $$\frac{a}{b}$$, where $$a$$ and $$b$$ are coprime positive integers. Find $$a+b$$.

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