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MATH 1B MIDTERM 2 (001)
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 (001)
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
n2 − 1
n3 + n+ 2
Solution:
(b) (15 points)∞!
n=1
n tan(1
n3)
Solution:
PLEASE TURN OVER
70m2 I L
2 n nim Lim I o
n n'tu 12u s nflo
L CTuh I
divergent1 u3tu 12u
Hospital
fan a L Sec2Gcim
aLim I
K 30 a o l 7 O
Lion tan LT Tu s
Cim Ataru I1 n I o
C C T uz Id n c
T 30Catarrh converg ent
n
Math 1B Midterm 2 (001), 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
nn
(2n)!
Solution:
PLEASE TURN OVER
nn
an I an I cn Inn2h 11 24 2
th1u 11 2
figs at e Liga la I o I
Ratio Test
T uL convergent
u
Math 1B Midterm 2 (001), Page 3 of 5
3. (25 points) Using only the integral test, determine whether the following series is abso-lutely convergent, conditionally convergent or divergent. If convergent you do not needto determine the sum.
∞!
n=1
"ln(n)
n
Solution:
PLEASE TURN OVER
positive Cts on Cl I
fix 1 T E tubal ta se 1 lula7 x2C
7121h x tt E Tulsa
71cal C o Fw all a eke z n
Can apply7cal decreasing an integral test
U tu Lal ddoc xd u
J da Jru du Fuk c
3 Kucxiphic
I da Lim du Eis ZenithA
Integral TEAso
e yetL
a divergent in I
Math 1B Midterm 2 (001), Page 4 of 5
4. (25 points) Determine the radius of convergence of the power series
∞!
n=1
(−1)nx2n
(4 + 1/n)n.
Solution:
PLEASE TURN OVER
2hu
X
au c nµ That 4th
fig Flat laterRoot Test LD 2h
T n abs uv it l tL t 4th T l
n i
s in 1 12T ya div it
47 I
1 4thu
txt 2
Radius at convergence 2
Math 1B Midterm 2 (001), Page 5 of 5
5. (a) (20 points) Calculate the Taylor series (centered at x = −1) of the function
f(x) = (x2 + 2x+ 1) sin(x+ 1)
Make sure to write a general term of the series.
Solution:
(b) (5 points) Using part (a), or otherwise, calculate f (2n+1)(−1), where n is a positivewhole number. You do not need to simplify your answer.
Solution:
END OF EXAM
C a
sina.ci K Epi E n2n I
Siu x 111 TyI 1 zu il C so I
n Ic a 1 Isit 2 1 sin x 117 x if sincxti 11
7
Taylor series at a I
Cza l D 712 c
n l I n 11 I
1 c i C it KuL it zu zu 11Zu 111
