Math 1220
Midterm 2
November 19, 2015
Please answer all the questions below. The value of every question is
indicated at the beginning of it. You may only use scratch paper and a
non-graphing calculator. No cell phones, notes, books or music players are
allowed during the exam.
Name:
UID:
1. (20 pts) Compute the following limits. If you use l’Hppital’s rule, please indicate exactly
where you are using it.
(a) (5 pts)
Rx√
lim
0
x→0
t cos t dt
x2
(b) (5 pts)
lim (sin x)cot x
x→0
(c) (5 pts)
lim x1/x
x→∞
(d) (5 pts) Explain why l’Hppital’s rule doesn’t work and compute the limit using a
different method.
x2 sin x1
lim
x→0 sin x
2. (10 pts) For what values of p does the following integral converge?
Z ∞
1
dx
xp
1
3. (10 pts) Determine whether the following improper integral converges or not
Z ∞
1
dx
2
x ln(x + 1)
1
Hint: Use a comparison theorem
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4. (15 pts) Are the following series divergent, conditionally convergent or absolutely convergent? Please indicate which convergence test you are using.
(a) (5 pts)
X (−1)n
1 + ln n
n≥1
(b) (5 pts)
X √n
n2 + 7
n≥1
(c) (5 pts)
X
n≥1
n
3n + 2
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n
5. (10 pts) Find the convergence set of the power series
X (−2)n+1 xn
n≥0
2n + 3
Make sure you study the behavior of the series at the endpoints of the convergence
interval.
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6. (10 pts) Find the power series in x through terms of degree 5 for the function
Z x −t
e
f (x) =
dt,
0<x<1
2
0 1+t
Remember that ex =
xn
n≥0 n!
P
and
1
1−x
=
P
n≥0
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xn .
7. (10 pts) Find a power series expansion for
1
(1+x)3
(a) (5 pts) By differentiating the power series
in 2 different ways:
1
1+x
=
P
n n
n≥0 (−1) x .
(b) (5 pts) By using the binomial series expansion (1 + x)p = 1 +
(c) (Extra credit - 5pts) Justify that both answers coincide.
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P
p
n≥1 n
xn
8. (a) (5 pts) Find the Taylor polynomial of order 2 for the function f (x) = ln[cos2 (x)]
around the point π6 .
(b) (5 pts) Give an upper bound for the error that we would make if we used the above
polynomial to approximate f ( π3 ). It suffices to write down the final expression that
you would plug into your calculator.
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9. (Extra credit - 10 pts)
(a) If we want to use a Taylor series in order to approximate sin(32), is it better to
expand f (x) = sin(x) around π6 or π3 ? Why?
(b) Taylor’s formula provides the error that is made when approximating a function by
its n-th Taylor polynomial
f (x) =
n
X
f k) (a)
k=0
k!
(x − a)k + Rn (x),
Rn (x) =
f n+1) (c)
(x − a)n+1 ,
(n + 1)!
c ∈ [x, a]
State the theorem that gives rise to the undetermined point c.
(c) Give an example of a series that converges conditionally (and justify your answer).
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