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 =

2) A function is given byh(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 =

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

M =

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

Recall that

2) A function is given by

3) Let

4) Does

You can earn partial credit on this problem.