Limits of Sequences and Series

Limits of Sequences: Level 2 Challenges


Tetration is defined as

na=aaan a’s.\Large {^{n}a} = \underbrace{a^{a^{\cdot^{\cdot^{a}}}}}_{n \ a\text{'s}}.

Find the value of

limnn(2).\lim_{n\rightarrow\infty}{^{n}}\big(\sqrt {2}\big).

Consider the sequence a1,a2,a_1,a_2,\ldots where an=cos(πn2+n)a_n = \cos(π\sqrt{n^{2}+n}). Find limnan.\large \lim_{n\rightarrow\infty} a_n.

Consider the sequence {an}\left\{a_n\right\} where an=(2x15)n.a_n=\left(\frac{2x-1}{5}\right)^n. What is the sum of all integers xx for which this sequence converges?

Let PnP_{n} be the product of the numbers in the nnth row of Pascal's Triangle.
limnPn1Pn+1Pn2=\lim_{n\to\infty}\frac{P_{n-1} P_{n+1}}{P_{n}^2} =

The sequence {an}\{a_n\} follows the recursion an+12=2an+3a^2_{n+1}=2a_n+3 with a1=7.a_1=7.

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


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