Limits of Sequences and Series

Limits of Sequences: Level 4 Challenges


limntan(89.99999999n 9’s)tan(89.99999999(n1) 9’s)= ? \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 an=an1an2an33a_{n}=\sqrt[3]{a_{n-1}a_{n-2}a_{n-3}} with initial terms a0=1,a1=2,a2=1a_{0}=1,a_{1}=2,a_{2}=1.

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


12×12+1212×12+1212+1212 \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 {an}\{ {a}_{n}\} satisfies (2n+12n1)an=e2. \left(\frac{2n+1}{2n-1}\right)^{a_n} =e^2. Find limn2ann+1 \lim_{n \rightarrow \infty}\frac{2a_n}{n+1} .

The sequence 1,2,3,2,1, 2, 3, 2, \ldots has the property that every fourth term is the average of the previous three terms. That is, the sequence {an}\{a_n\} is defined as a1=1,a2=2,a3=3a_1 = 1, a_2 = 2, a_3 = 3, and an+3=an+2+an+1+an3a_{n+3} = \dfrac{a_{n+2} + a_{n+1} + a_n}{3} for any positive integer nn.

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


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