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**Jhevon** Let $\displaystyle S_n = 3 + \frac {5}{2} + \frac {7}{3} + \frac {9}{4} + ...$

then $\displaystyle S_n = \sum_{n = 1}^{ \infty} \frac {2n + 1}{n}$

note that $\displaystyle \lim_{n \to \infty} S_n = 2 \neq 0$

thus the series diverges by the test for divergence

here is what the answer to the limit tells you. if the answer is zero then the series MIGHT converge, if it isn't zero, then it definitely does not converge

**Theorem: **Let $\displaystyle S_n$ be a sequence. If $\displaystyle \sum S_n$ converges, then $\displaystyle \lim_{n \to \infty} S_n = 0$

Note, this is an implication, the converse is not always true