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We have a recursive sequence $y_{n+1} = \sqrt{\frac{n+3}{n+1}} y_{n}$

for which value of $y_0$ does it converge?

I found this question in a set of Differential Equation problems. I don't have any idea how this can be solved using differential equations. The answer is probably only $0$ but I don't know how to solve this using Differential Equations.

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This recurrence relation can be solved with standard techniques. The result is $$y_n = \sqrt{\frac{(n+1)(n+2)}{2}}y_0,$$ which clearly diverges for any $y_0\ne 0$.

The connection to differential equations is the following: Rewrite the recurrence as $$y_{n+1}-y_n = \left(\sqrt{\frac{n+3}{n+1}}-1\right)y_n.$$ This corresponds to the differential equation $$y'(x) = \left(\sqrt{\frac{x+3}{x+1}}-1\right)y(x).$$ This can also be solved explicitly. The result is somewhat complicated but for large $x$ we find $y(x)\simeq c y_0 x,$ where $c$ is a (calculable) nonzero real number. This can be easily motivated. For large $x$ we have $$y'(x) \simeq \frac{1}{x}y(x),$$ and so $y(x)\simeq \mathrm{(const)}y_0 x$. More about the connection between recurrence relations and differential equations can be found here.

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$$\begin{align} y_{n+1} &=y_{n}\sqrt{\frac{n+3}{n+1}}\\ &=y_n\sqrt{1+\frac{2}{n+1}}\\ \end{align}$$ So we can find $\lim_{n\to\infty}y_n$ by writing $$\lim_{n\to\infty}y_n = y_0\prod_{n=1}^\infty \sqrt{1+\frac2{n+1}}=y_0\sqrt{\prod_{n=1}^\infty \left(1+\frac2{n+1}\right)}$$ But the product $$\prod_{n=1}^\infty \left(1+\frac2{n+1}\right)=\prod_{n=1}^\infty \left(\frac{n+3}{n+1}\right)=\left(\frac42\right)\left(\frac53\right)\left(\frac64\right)\left(\frac75\right)\dots=\frac{(n+2)(n+3)}6\to\infty$$ So one can conclude that $$\lim_{n\to\infty}y_n= \begin{cases} \infty &y_0\gt0\\ -\infty &y_0\lt0\\ 0&y_0=0 \end{cases}$$

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  • $\begingroup$ The problem is that the question was in A differential equation problem set. Most of the questions of this set were about first-order ordinary differential equations and their different forms like complete form , etc... And at the middle of them, there was this question. So I think It must have a solution using differential equations. $\endgroup$ – amir na Apr 14 '19 at 21:44
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    $\begingroup$ Sorry, I can't see an obvious solution by using differential equations to solve this. $\endgroup$ – Peter Foreman Apr 14 '19 at 21:45
  • $\begingroup$ Thanks anyway. If no one submit a solution with differential equations for 2 days, I will accept your answer. $\endgroup$ – amir na Apr 14 '19 at 21:47

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