Integral Test Quiz

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1. Does $\sum_{n=1}^\infty n^2e^{-n^3}$ converge or diverge?

   (a) Divergent because $\int_1^\infty x^2e^{-x^3} \;dx = \infty$
   (b) Convergent because $\int_1^\infty x^2e^{-x^3} \;dx < \infty$
   (c) Divergent $p$-series with $p > 1$
   (d) Convergent $p$-series with $p > 1$

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2. Does $\sum_{n=1}^\infty 5n^{-1/e}$ converge or diverge?

   (a) Divergent $p$-series with $p > 1$
   (b) Divergent $p$-series with $p \leq 1$
   (c) Convergent sum of two convergent $p$-series
   (d) Convergent $p$-series with $p > 1$

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3. Does $\sum_{n=1}^\infty \dfrac{2}{n\left(\ln{\left(5n\right)}\right)^4}$ converge or diverge?

   (a) Convergent $p$-series with $p > 1$
   (b) Divergent $p$-series with $p \leq 1$
   (c) Convergent because $\int_1^\infty \frac{2}{x(\ln{(5x)})^4} \;dx < \infty$
   (d) Divergent because $\int_1^\infty \frac{2}{x(\ln{(5x)})^4} \;dx = \infty$

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4. Does $\sum_{n=1}^\infty \dfrac{e^{1/n}}{n^2}$ converge or diverge?

   (a) Divergent because $\int_1^\infty \frac{e^{1/x}}{x^2} \;dx = \infty$
   (b) Divergent $p$-series with $p > 1$
   (c) Convergent because $\int_1^\infty \frac{e^{1/x}}{x^2} \;dx < \infty$
   (d) Convergent $p$-series with $p > 1$

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5. Is $\sum_{n=3}^\infty \dfrac{1}{n\ln{(n)}\ln{(\ln{(n)})}}$ convergent or divergent?

   (a) Convergent $p$-series with $p > 1$
   (b) Divergent $p$-series with $p \leq 1$
   (c) Convergent because $\int_3^\infty \frac{dx}{x\ln{x}\ln{\ln{x}}} < \infty$
   (d) Divergent because $\int_3^\infty \frac{dx}{x\ln{x}\ln{\ln{x}}} = \infty$

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