Practice Exam 1

Practice Exam 1
Made on 20120417. Distributed in class on 20120417. Made available online on 20120423.

#1) Evaluate the integral.
\[
\int\frac{6x+1}{(x+1)(2x-1)}dx
\]

#2) Find the area of the region enclosed by one loop of the curve.
\[
r=3\sin\left(3\theta\right)
\]


#3) Determine whether the integral is convergent or divergent. If
it is convergent, evaluate it.
\[
\int_{e}^{\infty}\frac{1}{x\left(\ln x\right)^{2}}dx
\]


#4) Determine whether the following are convergent or divergent.
State the type of test you use. You may use each type of test only
once.

a)
\[
\sum_{n=2}^{\infty}2\frac{3^{n}}{4^{n-1}}
\]

b)
\[
\sum_{n=2}^{\infty}\frac{\sin\left(n\pi\right)}{\log n}
\]
20120426: Changed the starting term from $1$ to $2$.

B)
\[
\sum_{n=2}^{\infty}\frac{\cos\left(n\pi\right)}{\log n}
\]
20120426: Added this problem, because the original b) was somewhat trivial.

c)
\[
\sum_{n=1}^{\infty}\frac{n^{2}}{n+1}
\]


d)
\[
\sum_{n=1}^{\infty}\frac{n^{n}}{\left(2+n^{2}\right)^{n}}
\]


e)
\[
\sum_{n=1}^{\infty}\frac{1+2n}{3n^{3}}
\]


#5) There should be three forms/strategies from section 11.7 which
you didn't use. State them.