Instructions: Your task is to complete the following problems. You must show your work for full credit!
Make sure to box your answers so I know where to find them and make sure to READ all the instructions
on each of the problems. May the odds be ever in your favor!
1. Answer the following questions about series. Make sure to explain all your answers in your own words.
(a) What does it mean for a series to converge?
(b) State all of the conditions of the regular comparison test. Then give an example of a series that
fails the regular comparison test.
(c) Give an example of a series that fails the ratio test.
2. Determine if the following series converge or diverge. Make sure to state the test you use and justify
your answers for full credit.
(a) P∞
n=1
1
(n2+1)e
tan−1(n)
(b) P∞
n=1(−1)ne
3/n
(c) P∞
n=1
ncos2
(n)
e+n5
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(d) P∞
n=2
(−1)n−1n
n
(lnn)n
3. Determine if the series converges absolutely, conditionally, or diverges: P∞
n=1
(−1)n(n+2)
n
√3 n
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4. Hey! Here is the Maclaurin series for tan−1
(x) = P∞
n=0
(−1)nx
2n+1
2n+1
(a) Use the Maclaurin series above to find a Maclaurin series for f(x) = x
2
tan−1
(4x
2
)
(b) Use the Maclaurin series from (a) that you obtain to evaluate the integral: R
x
2
tan−1
(4x
2
) dx
(Hint: Your answer should be another series. Simplifying before you create the series may get you
in trouble…)
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5. Solve the following differential equations.
(a) (cos2x)y
0 = e
y+tanx
(b) x
2y
0 − y = 2x
3
e
−1/x
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6. Find the radius and interval of convergence.
(a) P∞
n=1
2
n(x+6)n
(n+2)!
(b) P∞
n=1
4
n(x−2)n
√
n3+3
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7. Find a Taylor series for f(x) = 5
(2x+1)2 at a=1. (Hint: Try not to simplify too much… or you could lose
your pattern!)
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8. Arina has hopes and dreams of preparing a ton of margaritas for her post Calculus 2 party. She wants
to try and make a blood orange infusion and mix it into her Margaritas. She has a tank that is filled
up with 100L of pure vodka. Vodka with 0.6 kg/L of blood orange infusion concentration is added at a
rate of 8L per minute. The solution is kept mixed and drained from the tank at a rate of 4L per minute.
Find the amount of blood orange infusion in the tank after 12 minutes.