# Tagged Questions

Questions regarding functions defined recursively, such as the Fibonacci sequence.

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### Frobenius Method

We have been given a Hermite equation $\frac{d^2 y}{dx^2} -2x \frac{dy}{dx}+2ny=0$ We need to use the Frobenius method to solve. So far we have solved the indicial equation and got r = 0,1 and the ...
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### Recurrent sequence limit

Let $a_n$ be a sequence defined: $a_1=3; a_{n+1}=a_n^2-2$ We must find the limit: $$\lim_{n\to\infty}\frac{a_n}{a_1a_2...a_{n-1}}$$ My attempt The sequence is increasing and does not have an upper ...
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### If $s(1)=1$ and $s(n)=s(n-1)+\text{lcm}[n,s(n-1)]-(-1)^n$ then $\lim\limits_{n\to \infty}\frac{s(n)}{(n+1)!}=\frac{1}{e}$

Define the sequence $(s(n))$ recursively by $s(1)=1$ and, for every $n\ge2$, $$s(n)=s(n-1)+\text{lcm}[n,s(n-1)]-(-1)^n.$$ Prove that $$\lim_{n\to \infty}\frac{s(n)}{(n+1)!}=\frac{1}{e}$$ I got ...
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### How to find the first 5 values of a recursive relation where certain sequences are not known?

Write down the first five values of each of the following recursive sequences. (a) r(0) = 2, r(n) = [r(n-1)] -n -1 for all integers n>=1 (I couldn't write the values as ace of r and s so I just wrote ...
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### Periodicity of solutions of rational difference equations $x_{n+1}=\alpha+\frac{x_{n-1}}{x_n}$ [closed]

\begin{equation*}x_{n+1}=\alpha+\frac{x_{n-1}}{x_n}\tag{1}\end{equation*} Equation(1) has solutions of prime period 2 if and only if $\alpha = 1$. How to prove this? Thanks.
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### Find $\sqrt{4+\sqrt[3]{4+\sqrt[4]{4+\sqrt[5]{4+…}}}}$
Find the value of $$\sqrt{4+\sqrt[3]{4+\sqrt[4]{4+\sqrt[5]{4+...}}}}$$ I know how to solve when all surds are of the same order, but what if they are different? Technically, (as some users wanted ...
The assignment is the following: (a) Given a sequence $(a_n)_n$ which satisfies the recursive equation $a_n = \sum\limits_{k = 1}^d c_k \cdot a_{n-k}$ with $c_d \not= 0$. Furthermore \$Q = 1 - c_1t - \...