Post on 18-Jul-2020
MATH 1B MIDTERM 2 (PRACTICE 1)
PROFESSOR PAULIN
DO NOT TURN OVER UNTILINSTRUCTED TO DO SO.
CALCULATORS ARE NOT PERMITTED
THIS EXAM WILL BE ELECTRONICALLYSCANNED. MAKE SURE YOU WRITE ALLSOLUTIONS IN THE SPACES PROVIDED.YOU MAY WRITE SOLUTIONS ON THEBLANK PAGE AT THE BACK BUT BESURE TO CLEARLY LABEL THEM
ex = 1 + x+ x2
2 + x3
6 + · · · =!∞
n=0xn
n!
sin x = x− x3
3! +x5
5! −x7
7! +x9
9! − · · · =!∞
n=0(−1)n x2n+1
(2n+1)!
cos x = 1− x2
2! +x4
4! −x6
6! +x8
8! − · · · =!∞
n=0(−1)n x2n
(2n)!
arctan x = x− x3
3 + x5
5 − x7
7 + x9
9 − · · · =!∞
n=0(−1)nx2n+1
2n+1
ln(1 + x) = x− x2
2 + x3
3 − x4
4 + x5
5 − · · · =!∞
n=1(−1)n−1xn
n
(1 + x)k = 1 + kx+ k(k−1)2! x+ k(k−1)(k−2)
3! + · · · =!∞
n=0
"kn
#xn
limn→∞(n+1n )n = e
Name:
Student ID:
GSI’s name:
Math 1B Midterm 2 (Practice 1)
This exam consists of 5 questions. Answer the questions in thespaces provided.
1. Determine if the following series converge or diverge. If convergent you do not need togive the sum. Carefully justify your answers.
(a) (10 points)∞!
n=1
sin(πn
2n+ 1)
Solution:
(b) (15 points)∞!
n=1
√n
n+ 1
Solution:
PLEASE TURN OVER
Sin continuousd I
Zu
TuLim Iz Liz sin II I sin Iz l toh
sin drift by divergence test
rnlet an bu fu an bn 0 For all u
Lim anu Tu fig I o
C C TLT T TuL divergent L divergentu i
y u i
pseries p C l
Math 1B Midterm 2 (Practice 1), Page 2 of 5
2. (25 points) Determine if the following series is convergent or divergent. If convergentyou do not need to determine the sum.
∞!
n=1
1 · 4 · 7 · · · (3n− 2)
(2n)!
Solution:
PLEASE TURN OVER
I 4 7 3 u zan
n
I 4 7 3h Z 3 uti 21antn Gutilczu 12
anti 3h I1 1
zun tie laa I o l
T l 4 7 3 u zconvergentGull
u
Ratio Test
Math 1B Midterm 2 (Practice 1), Page 3 of 5
3. (25 points) Using the integral test, determine whether the following series is absolutelyconvergent, conditionally convergent or divergent. If convergent you do not need todetermine the sum.
∞!
n=1
(−1)nn
en
Solution:
PLEASE TURN OVER
3
Math 1B Midterm 2 (Practice 1), Page 4 of 5
4. (25 points) Determine the domain of the function f(x) given by the power series
∞!
n=1
(−1)n(2x− 4)n
n3/2.
Solution:
PLEASE TURN OVER
Math 1B Midterm 2 (Practice 1), Page 5 of 5
5. (a) (20 points) Calculate the Maclaurin series of the function
f(x) =1√
4 + x2.
You need only write out the first four terms. You do not need to simplify thecoefficients.
Solution:
(b) (5 points) What is the radius of convergence of this Maclaurin series?
Solution:
END OF EXAM
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Binomial Expansion
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R O C 2