# Series practice

When answering the following, keep in mind series and series tests that we have discussed …

• nth term divergence test
• telescoping
• geometric
• integral test
• p-series
• direct comparison
• limit comparison
• alternating series test
• absolute convergence
• root test
• ratio test

Determine convergence or divergence of the following series.

1. $$\displaystyle\sum_{n=1}^{\infty} 2^{1/n}$$

2. $$\displaystyle\sum_{n=1}^{\infty} \dfrac{n!}{(2n)!}$$

3. $$\displaystyle\sum_{n=1}^{\infty} \dfrac{ (-1)^{n-1} }{\sqrt{n}}$$

4. $$\displaystyle\sum_{n=2}^{\infty} \dfrac{1}{n (\ln n)^3}$$

5. $$\displaystyle\sum_{n=1}^{\infty} 4^{-2n+1}$$

• geometric series with $$r = \dfrac{1}{16}, \left| \dfrac{1}{16} \right| \lt 1$$, so series converges
(note: the sum of the geometric series is $$\dfrac{4}{15}$$ )
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