Calculus 2 test

Math3B Exam #03

Spring 2021 Solution 1. Determine if the sequence converges, and if so find the limit.

a) ( ) ( )

1

2 1 ! 2 1 !

n

n n

∞

=

− +

b) 1

1 2

n

n

n n

∞

=

+ +

fin.an kn.IE ii k an k In Let g htt

IznFznt lny nlnlh fEoI4nEI Lindens fi.no

nt

D Convergent yjln

tin IX x 2 I

L'm 1 x 2 I X soo I

X

Elin X soo

1 z

x

o I to

I

I finally I Ln'Iooy E

CONVERGENT

2. Find the smallest value of n for which the nth partial sum is ensured to approximate the sum of the series to the

stated accuracy. Recall: 1n ns s b +− ≤

( ) ( )( )11

1 ; 0.001

1 3

n

n n

error n

∞

− =

− <

+ ∑

WE know S sa E bate

WE Require bn e 0.001 7

bn nie 3 I

7 butI Chtz 3h

bn e 0.001 I

tz 3h 0.001

I tz zn

I 1000

ht 2 3h 1000

h 3 5 33 5.27 135

n 4 G 34 6.81 486

n 5 7.35 7 243 1,701

for n 5 ERROR 20.00 I

3. Use the indicated test to determine if the series converges or diverges.

a) 22

3

1 2

n

n

n n

∞

=

− −

∑ Root Test b) 1

! n

n

n n

∞

= ∑ Ratio Test

firsoonian find IT fine aan't htt htt

L Enn figo.fmyfe.hIn I

Ey Enn

lny enenCE.tn fI oIhIa

n

fi 6Dfi.no n'II's l nail k Ih I fin.IN

n

I

o.IE EEI.EEiEtM fia Iten n

fim IEIK.IE I fiaoo ce

i I

Einsoolet In

Ee I L z 251 1

Nnn is Conveesert Z

a finally 2

fizzy e e

n Enn is Diversent

4. Find the interval of convergence for the series below. Then find the sum of the series.

( ) ( ) ( ) ( )2 3 41 2 1 2 1 2 1 2 1x x x x− − − −− + − − − + − − − + "

5. Match a series from A), B), C), D) with the graph of its sequence of partial sums a), b), c), d).

i. 1

1

3 4

k

k

−∞

=

−∑

ii. ( ) 1

1 1

7 1

6 k

k k

∞ −

− =

−∑

iii. ( )( )1

1 2 3k k k

∞

= + + ∑

iv. 2

1

1 9 3 2k k k

∞

= + − ∑

This is a geometric SERIES with a 1 and r 2 1 FOR CONVERGENCE WE REQUIRE A5 I rr I

Ex e 2 I I I S

e 2 I I I I CE e5 O 2 2

S I

0C 2 It 2 1

00 2 I Iz S z x Iz 2X c 00

legal ex 00 5 2 1 X OO

Interval of CONVERGENCE 1,1 00

5 1,52 14,53 6 L tEzt b Ii i

Et a

Et c iii iv

Iot d

6. Find a closed form for the nth partial sum ( ns ) of the series below. Be sure to take into account that the index k

does not begin at one. Note: You don’t need to find the sum of the series.

a) 2 3

1 1k k

∞

= − ∑

b) ( )2 1

1k k k k k

∞

=

+ −

+ ∑

E III Ice Ee t 7 A k e t B late

K L I 2B B I

sn Ite Ie k e e za A I

Sn I IIe IIe Kyte kentzI Sn E E I If t t f t t

Hei n's

sa Ez E t F NII nts

EEnEEIirEiE.ii EE.E ii.snIEI III Ken Kent

t I

sn E E t.r.tl nH tE

Sn TE ni

7. Use the limit comparison test to show that the series 1

1 1 ln ln

k e

e k

∞

=

+ +∑ converges or diverges.

Choosing It Divergent Harmonic SERIES

s.la fi.n.IlnltetIaltI e k O

l latte Is

fine ee't

Eino ee't I I

E

C Finite

I.IE ln Ettz tlneisDiueesEut

8. Classify the series as absolutely convergent, conditionally convergent, or divergent.

( ) ( ) ( )

2

3 1

1 1

2

n

n

n

n

∞

=

− +

+ ∑

Checking 2 an Checking Zan

ant ECEI an E Iet.cn 5 Using the LCT with Eth Using the AST divergent harmonic SERIES Let fix I X Z I

1 Abi f'cµ 2 63 2 Crete 3 2

X3 2 2

Liz Inn's'IE f 2x4t4x 3 4 3 2

43 2

fins thin fl x 3 2 4x

3 2 2

L Finite f X x3t3xt4

X3 2 z O

E an is divergent X Z I

bn Zbinter

III bn k In O

I Zan is CONVERGENT

7 1 ritzh3 2 is ConditionallyConvergent

9. Find the radius of convergence and the interval of convergence. Determine if the series is absolutely

convergent, conditionally convergent, or divergent at the left endpoint of the interval. Be sure to show your

work.

( ) 1

3 2

k

k k

x k

∞

=

− ⋅

∑

Using the Ratio TESE

fin 6kt Checking X L the left Endpoint

Ak of the interval of CONVERGENCE K

a si EEE3 E.ie

fim cxss Ee.iz EEitEEdk EEceY

E I 7 k

x s.E fin.lk ask I Iz X 3

Checking Zlak FORabsolute CONVERGENCE 2 laid DivergentHaemonic

SERIESWE REQUIRE Iz X 3 I

I E x 3 e e r 2 laid is divergent

Z X 3 c 2 Checking Zak

1 x 5 Zak III If I 5 Using the A

ST

Let fCxS X z I

f a 12 0 XII

Interval of CONVERGENCE bkzbkt.ee Kz I

II 5 fins.be fim o or

R I TEI 23 is conditionallyconversen

10. Find the Maclaurin Series for ( ) 1tanf x x−= . Express your answer in sigma notation.

Note: You must clearly show your work to get credit for this problem!

WE know f 17 2 dx tan k t c 7 I

7 t Xz I C z I XZ t X't X G t XD

This is a geometric series with a L and r L K rl x2 I

I X I

f zdx tan x t C Ke x2tx4 xGtx8 d

tan x C X t t t Cz 00

tan x Z y n X

t

h L 2h te t C

since tan 0 0 then C D

tan x n X2ate

anti

11. Use the first three terms of the Maclaurin Series for ( ) ( ) 1 1

ln 1 1 k

k

k

x x

k

∞ +

=

+ = −∑ to find the x-coordinate(s) of

intersection(s) of the graphs of ( ) 3( ) ln 1 ( )f x x and g x x= + = .

gcxI x Y

lax ex t fcxt lnle.tl

Setting x I s

66 1 51 696 e trios x

6 3 2 2 3 6 3

0 4 3 3 2 Cox v

0 144 2 3 6 r v

4 0 442 3 6 0

3 I 32 414 6 214

i

X 31 19 96

8

X 3 1 cos 8