I have the sequence $$X_n \rightarrow \frac{n^2}{n^2+31n+228}$$ With limit $X_0 = 1$ and I want to use an appropriate conditional statement in Maple $17$ to find N such that $\lvert Xn-X_0\rvert < \epsilon$ for every $n\ge N$ and I need to produce an appropriate list of data points $(n,X_n)$ to illustrate this.
I'm struggling for days now to solve this question, if anyone knows the solution please answer as soon as possible.
Let $1-\epsilon=\frac1{1+\epsilon_0}$. Clearly, $X_n<1$ for every $n$. So, we need to guarantee $X_n>1-\epsilon=\frac1{1+\epsilon_0}$, i.e. $\epsilon_0n^2-31n-228\ge 0$. This polynomial goes to $\infty$ as $n\to\infty$, because, the leading coefficient is positive. So, it is enough to choose $N=\frac{31+\sqrt{31^2-4\cdot228\cdot\epsilon_0}}{2\epsilon_0}$, which is the maximal root of $\epsilon_0n^2-31n-228$.