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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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