Question

Sol. [Key Concept: Simplify the sum over $n$ to a closed form.]

$${n^{2} < n^{2} + 1 < n^{2} + 2 < \ldots < n^{2} + n }{\Rightarrow \ \frac{1}{n^{2} + 1} + \frac{2}{n^{2} + 2} + \ldots + \frac{n}{n^{2} + n} < \frac{1}{n^{2}} + \frac{2}{n^{2}} + \ldots\frac{n}{n^{2}} = \frac{1}{2} + \frac{1}{2n} }$$and

$$\frac{1}{n^{2} + 1} + \frac{2}{n^{2} + 2} + \ldots + \frac{n}{n^{2} + n} > \frac{1}{n^{2} + n} + \frac{2}{n^{2} + n} + \ldots\frac{n}{n^{2} + n} = \frac{1}{2}$$

✔ (a) $S < \frac{12 +}{\frac{1}{2n}}$

✘ (b) $S < \frac{12 +}{\frac{1}{2n}}$

✔ (c) $S > \frac{12}{}$

✔ (d) $S < 1$