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Question

Use the Integral Test to determine the convergence or divergence of the p-series. n=11n0.9\sum_{n=1}^{\infty} \frac{1}{n^{0.9}}

Solution

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The function f(x)=1x0.9f(x)=\dfrac{1}{x^{0.9}} is positive, continuous and decreasing for x1x\geq 1. Therefore, ff satisfies the conditions for the Integral Test.

11x0.9dx=1x0.9dx=limb1bx0.9dx=limb[10x0.1]1b=limb[10b0.110(1)0.1]=\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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