급수 수렴 조건과 수열의 극한

$\displaystyle \sum\limits _{n=1} ^{\infty} \left( na _{n} - \frac{n ^{2} + 1}{2 n + 1} \right) = 3$이므로 $\displaystyle \lim\limits _{n \rightarrow \infty} \left( na _{n} - \frac{n ^{2} + 1}{2 n + 1} \right) = 0$이다. 즉, $\displaystyle \lim\limits _{n \rightarrow \infty} n \left( a _{n} - \frac{n + \frac{1}{n}}{2 n + 1} \right) = 0$에서 $\displaystyle \lim\limits _{n \rightarrow \infty} n \neq 0$이므로 $\displaystyle \lim\limits _{n \rightarrow \infty} \left( a _{n} - \frac{n + \frac{1}{n}}{2 n + 1} \right) = 0$ $\displaystyle \therefore \lim\limits _{n \rightarrow \infty} {a _{n}} = \lim\limits _{n \rightarrow \infty} {\left( a _{n} - \frac{n + \frac{1}{n}}{2 n + 1} \right)} + \lim\limits _{n \rightarrow \infty} {\frac{n + \frac{1}{n}}{2 n + 1}}$ $\displaystyle = 0 + \frac{1}{2} = \frac{1}{2}$ $\displaystyle \therefore \lim\limits _{n \rightarrow \infty} {( a _{n} ^{2} + 2 a _{n} + 2 )}$$\displaystyle = \left( \lim\limits _{n \rightarrow \infty} {a _{n}} \right) ^{2} + 2 \lim\limits _{n \rightarrow \infty} {a _{n}} + 2$ $\displaystyle = \left( \frac{1}{2} \right) ^{2} + 2 \times \frac{1}{2} + 2 = \frac{13}{4}$