Limit of a Sequence 06

" $\displaystyle{\sum_{n=1}^{\infty}a_n}$ diverges"; By the ratio test, if $\displaystyle{\lim_{n \rightarrow \infty} \left|\frac{a_{n+1}}{a_n}\right| < 1}$ , then $\displaystyle{\sum_{n=1}^{\infty}a_n}$ converges.

" $\displaystyle{\sum_{n=1}^{\infty}a_n}$ converges to a positive finite value"; Let $a_1 = -1$ , and $\displaystyle{a_n = \frac{1}{2^n}}$ for all $n \geq 2$ . Then $\displaystyle{\lim_{n \rightarrow \infty} \frac{a_{n+1}}{a_n} = \lim_{n \rightarrow \infty} \frac{\frac{1}{2^{n+1}}}{\frac{1}{2^n}} = \lim_{n \rightarrow \infty} \frac{1}{2} = \frac{1}{2} < 1}$ , but $\displaystyle{\sum_{n=1}^{\infty}a_n = -1 + \sum_{n=2}^{\infty}\frac{1}{2^n} = -1 + \frac{1}{2} = -\frac{1}{2}}$ .