Chapter 8: Infinite Sequences and Series

Section 8.3: Convergence Tests

Example 8.3.22

$$ Determine if the series $\sum _{n\=1}^{\infty }\frac{{\left(-1\right)}^{n\+1}}{n \sqrt{n}}$ diverges, converges absolutely, or converges conditionally. If it converges conditionally, determine if it also converge absolutely. $$ Solution $$ Since the series is alternating, it is a candidate for the Leibniz test. Now $a_n\=1\/n^{3\/2}$, which generates a sequence that is monotone decreasing to zero as $n\to \infty$. Hence, by this test, the series converges conditionally.   Moreover, since $\mathrm{\Σ} a_n$ is a p-series with $p\=3\/2\>1$, the series converges absolutely.$$ $$

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