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Question

# Use the Integral Test to determine the convergence or divergence of the p-series. $\sum_{n=1}^{\infty} \frac{1}{n^{0.9}}$

Solution

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The function $f(x)=\dfrac{1}{x^{0.9}}$ is positive, continuous and decreasing for $x\geq 1$. Therefore, $f$ satisfies the conditions for the Integral Test.

\begin{align*} \int_1^\infty \dfrac{1}{x^{0.9}} \, dx &= \int_1^\infty x^{-0.9}\, dx \\ &= \lim\limits_{b \to \infty} \int_1^b x^{-0.9}\, dx \\ &= \lim\limits_{b \to \infty} \left[ 10x^{0.1} \right]_1^b \\ &= \lim\limits_{b \to \infty} \left[ 10b^{0.1} - 10(1)^{0.1}\right] \\ &=\infty \end{align*}

So, the series diverges.

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