1) Suppose that $f(x)$ is a function that is positive and decreasing. Recall that by the integral test:
$$\displaystyle \int_p^{\infty} f(x)\, dx \leq \sum_{n=p} ^{\infty} f(n).$$
Recall that $\displaystyle e = \sum_{n=0}^{\infty} \frac{1}{n!}.$ Suppose that for each positive integer $k$, $f(k) = \frac{1}{k!}$. Find an upper bound $B$ for $\displaystyle \int_2^{\infty} f(x)\, dx.$
$B$ =

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2) A function is given by $$h(k) = \displaystyle \int_0^{\infty} x^k e^{-x} dx.$$ Its values may be found in tables. Make the change of variables $y = x \ln(4)$ to express $\displaystyle I = \int_0^{\infty} x^{2} 4^{-x}dx$ as a constant $C$ times $h(2).$ Find $C$.
$C$ =

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3) Let $g(x) = x^2 4 ^{-x}$. Find the smallest number $M$ such that the function $g(x)$ is decreasing for all $x > M.$
$M$ =

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4) Does $\displaystyle \sum_{n=1}^{\infty} n^{2 }4 ^{-n}$ converge or diverge? ? Converge Diverge