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Determine the convergence or divergence of the series using any appropriate test. Identify the...

Question:

Determine the convergence or divergence of the series using any appropriate test. Identify the test used.

{eq}\sum_{n=1}^{\infty} \frac{4}{3 \cdot \sqrt{n^3}} {/eq}

Answer and Explanation:

{eq}\text{We will apply the integral test.}\\ \int_{1}^{\infty}\frac{4}{3x^{\frac{3}{2}}}dx\\ =\int_{1}^{\infty}\frac{4}{3}x^{-\frac{3}{2}}dx\\ \text{Applying the formula:}\\ =\left [\frac{-8}{3\sqrt{x}} \right ]_{1}^{\infty}\\ =\frac{8}{3}\\ \text{it is a finite value so we say that the series converges.} {/eq}


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