Name:
Math 2300 Section 005 – Calculus II – Fall 2010
Quiz – Due Tuesday, October 12, 2010
This quiz is intended to be a review, so it contains more problems than usual. I won't be grading each problem individually, but the quiz will be graded as a whole. Solutions will be posted on Tuesday.

Find the sum
Hint: Write this as two geometric series.
You should be able to solve from here.

Do the following sums converge or diverge?

Hint: The first thing I think when I see this is that the integral
can be done using a substitution. I also notice that the function is positive. The derivative is given by
and setting this equal to zero shows that we have critical points where . You may want to convince yourself that this function is indeed decreasing beyond, say, . Then, consider convergence of the series

Hint: The obvious way to approach series with factorials is with the ratio test. I'll run through the limit calculation here and let you decide the result. You should know how to calculate with factorials.
It's up to you to do the calculation from here.

Hint: Generally, the first thing you want to do when you see an alternating series is to use the alternating series test. This series is clearly alternating and satisfies the property needed in the test. Can you explain why? What is the result of the test?

Hint: This one is a bit more random. First, since
shows that our function is bounded above by a divergent function, we don't get anything immediately from the comparison test. Maybe we can use the limit comparison test instead. Let
Then we find that
and I'll let you finish from here.

Hint: Notice that
and while it is clear that the derivative is negative (i.e., for the differences will be less and less) we find that
This is a tricky problem, but what you may want to try to do now is to relate this function to a series that we know diverges. I'll let you try to finish the next portion, or find a better way of doing the problem in general.


Find the radius of convergence of
This won't be on the exam. Try to complete it, but don't put it at such a priority as the other problems.

Find the interval of convergence of
This won't be on the exam. Try to complete it, but don't put it at such a priority as the other problems.