The first part is less specific than this.

jasperE3 wrote: Set $a_n=\frac{2n}{n^4+3n^2+4},n\in\mathbb N$. Prove that the sequence $S_n=a_1+a_2+\ldots+a_n$ is upperbounded and lowerbounded and find its limit as $n\to\infty$. Just write $a_n=b_n-b_{n+1}$ where $b_n=\frac 1{n^2-n+2}$ And so $S_n=b_1-b_{n+1}$ and so $\boxed{\lim_{n\to+\infty}S_n=b_1=\frac 12}$