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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 2

Tetration is defined as

$\Large {^{n}a} = \underbrace{a^{a^{\cdot^{\cdot^{a}}}}}_{n \ a\text{'s}}.$

Find the value of

$\lim_{n\rightarrow\infty}{^{n}}\left(\sqrt {2}\right).$

Consider the sequence $$a_1,a_2,\ldots$$ where $$a_n = \cos(π\sqrt{n^{2}+n})$$. Find $\large \lim_{n\rightarrow\infty} a_n.$

Consider the sequence $$\left\{a_n\right\}$$ where $$a_n=\left(\frac{2x-1}{5}\right)^n.$$ What is the sum of all integers $$x$$ for which this sequence converges?

Let $$P_{n}$$ be the product of the numbers in the $$n$$th row of Pascal's Triangle.
$\lim_{n\to\infty}\frac{P_{n-1} P_{n+1}}{P_{n}^2} =$

The sequence $$\{a_n\}$$ follows the recursion $$a^2_{n+1}=2a_n+3$$ with $$a_1=7.$$

Determine $$\displaystyle \lim_{n \to \infty} a_n$$.

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