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

Has I ItMaclaurin series of 7 x

i it 5 I i P

Binomial Expansion

I e i lay a 4 late 2

R O C 2