Integral Test Quiz

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1. Does $\sum_{n=1}^\infty \dfrac{\pi^2}{n^2 + 1}$ converge or diverge?

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

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

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

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3. Is $\sum_{n=1}^\infty ne^n$ convergent or divergent?

   (a) Divergent since $\int_1^\infty xe^x\;dx = \infty$
   (b) Divergent since $\lim_{n \to \infty} ne^n = \infty$
   (c) Convergent since $\int_1^\infty xe^x\;dx < \infty$
   (d) Convergent since $\lim_{n \to \infty} ne^n = 0$

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

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

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