These are two sequences we want to study convergent for it I need a hint to determine whether the following sequences convergent or divergent. $$\mathcal X_n=\frac{1+2+3+...+n}{n^2}$$ For this I have already tried $$\mathcal 1+2+3+...+n =\frac {n(n+1)}{2}$$ so $$\mathcal X_n=\frac{n+1}{2n}$$ hence$$ \mathcal X_n\to \frac{1}{2}$$. $$\mathcal Y_n =\frac{2.4.6...2n}{1.3.5...(2n-1)}$$ What should I use to solve it.
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Your conclusion for $X_n$ looks fine. For $Y_n$, a similar question has been answered here: To show for following sequence $\lim_{n \to \infty} a_n = 0$ where $a_n$ = $1.3.5 ... (2n-1)\over 2.4.6...(2n)$
