The sum of an infinite sequence of numbers

Are A t Az 1 A 3 t dy t

n Partial sum Sn is thesum of the first n terms

She y Ak Al ta z t tan

Sum of a series Effare sum diff Sn

Series converges if its sum a real diverges if its
sum D x or DN E Sum can often be impossible to find
however
Consider 7 I tty t f t
t
I E
Partial sums i s I
Sr
I I I I I
l f
S 3 Iz t te t f
Sy I
I 16
I t t t t To 6

Sn Ln

Take infinite limitto find sum Lima I

f I I I o I

A tar tartar 3 tar 4

GeometricSeries EE Ar
Multiplying by r each time first term is ALWAYS a
How to recognize a
geometric series
ITonlyappears in the power of numbers
K is only to the first power
2nd term
A the firstterm Plugin initial value of K F ist term
or

the number that'sbeing raisedto some kind of K power

A geometricseriesconverges if Irl al Gic ra t diverges

otherwise rel or r ZD If a geometric series converges

its sum is Er
El Dothe following series converge or diverge If they
converge findtheir sum
i
t l converges
EE 3LET
Fi Ey F
a 1st term 312 sum 3
3

OR REWRITE E 365 3 LEF E LEF

b
EE 35,11 Eee 3 41 EEEEd E Inverse
a I term
if I Ifs fr sum
77 T I's

c
EE EE
s a E EH a
E r 6

Diverges ble 621

2 converges
d EE EE EE 3 t a tz
sum Z 1
i t
TelescopingSeries Middle terms cancelling out

EZ Findthe sum of
a metric ble k is not a

II E Piter.es

Sn t E IEEE IJA ta
I I
sum him l E I I t o I

Use tests to determine whether a series converges diverges

Test for divergence
É an diverges if the
hit an
Test cannotbeused toshow convergence
0 where Ken

If tiny an o test is inconclusive doesnt tell us anything

Could conv or div we don't know
Integral Test Let EE be a series Let f n an for all
integers n z n such that f is a continuouspositive decreasing function
of X forall xz n Then the series Egan converges if andonly
if
If e dx converges Either both conv or both dir

E3 Do the following series conv or div t why State any

test used
A to diverges via the
EE 31 n'If 34nF 3q test

for divergence
2
0 1

b EE Ke xe time xe Ex du wax
do xd x
b

III II'eau him LET find fee ED

I
fee D I e
D If D I real

series converges via Integral test

3n The limit will

e FEED him I if him thnx
JUMP backandforth b w x t X Limit D NE t o diverges

properties of Series Onlyways tomanipulate series

Let Earn Ebre be convergent series t C be a constant
Such that Earn A Ebu B x
A B
AK
Sum Difference
QE que Ibn t.E.br

constant multiple
CEEare A
FE can
Neverpull K out of series Neverseparateby Matt Idivision
ONU I CON Conv
di v I conV dir
 div t dir div
As long as series is eventually convergent it is always
convergentGame fordivergent

f Series G Positive constant

converges for p I
7 he Diverges for o L P El

Comparison Tests F 2 lo
convergent e series
Consider l t
t tt tt t

I
1 Iz t's t fo t t

Each term in is less than the corresponding term

in
Ip Ie 7 nti nk corn e series
we can see
E In
so

FEE Ice Musthave a real sum convergent

Positive terms only

Direct ComparisonTest Let EaktEbk be series such that
C Are a be t
0 AK a bn and bra zone
IF Ebu converges then Eau also converges Gnu an

diverges then div Div

 7 Conv means nothing
E div means nothing
F I Do the following series conu or diverge
A l
I 2h I
l
zn i In
Tryto compare to the highest term on top highest term
on bottom if possible

In I t EEE IT I diverges K series

b Ei k ke ZE
ke
en
E
en f
E a con geometric series

n l n n
x K I L
ti E T
3155311 1 Inst S i3 53N I
K

2 Gnu 9 series
IT tha t
7
conv Direct comp Test
r

I less than I greaterthan

htt Intl
K't 3kt
ni I
I
d
Esri
I msn.sc In
EE Div P series so die 11Direct comp test
 bigger numerator biggerfraction
D Tft 3kt I n2 biggerdenominator smallerfraction
E SE peu w nu

If youget not obvious C Div or corn use the limit comp test

Limit Comparison Test

Let Ean t Ebk be series
I If diff Afn L t o Lord then either both Eau c Ebt
converge or both diverge

2 If hif.no barn o Ebt Con then Eau corn

3 If n'The
En re t
Eba div then Eau div
AMlore to K p series
k't 3kt I EE FE corn
e E
Kei Ku the
4

F2 Dothe following series conv or dive why Stakany tests

used

A 2 37 1 1 P series b c
7
AEF Ku 2 2 Compare to LE
I

im n2t3nH p2 n 43 htt
X
n 4th
nf n2tl
Oa Icao Satisfying conditions of limitcomptest

B c the compseries converges the original series also

converges through the limit comp test
b EE EEEifIa sdiu e serie
SEIIt come to EE.ir n

lion Sn l n Snk n't lime

ssomePowers
mo Ent NEE
o Sze Theoriginal series diverges
via the limit comparison test

81545kt I
C
2 3 y
compare to 1 3 1 Conu E series

innit
n m
him t.im

Samepowers lim Oa
fee
Both Series converge through limit comp test
Mlk I
link 1
A E compare to thento 1 Alak K
753 FEI WEI SE lak I fork e

MK I
div is div via direct come test

Zarctank I
Arctank L
E
b XEI 4
are tanks In
3 K t I

Zarctank 1 5 P series
a 2 Hz l
Kt 3 4 Ig
Corn Conu Original series convergesthrough
direct comp test
t E IIIs 347 fakes
a 34 com e series

Leon conv Series is conu via directcomp test

a Ink Ink 71 for ki e nottrue for firsttwo
do 2
I RFT terms but true forevery term after
e
fork
MI I div 7 series
ok ok
dive div series is div via directcomp test
Div P series
Khotan K'arctan
C
EI
div d a
series is div via directcomp test

Aseriesin which the terms are alternating positive negative is

called an alternating series C one alternating component
ex
EE th l
It's It's
Can never use integral test or either of the comp tests
notcontinuous positivetermsonly

The C Mak or
Alternating Series Test series EE
GEC1 ax cow if the following 3 conditions are satisfied
1 An O Ifonet condition fails
2 An is non increasing
lim An
O
decreasing
y the test is inconclusive
El Connor div why
a EE E t's
I An of
Z En Htt Z
o
Fn ntl ntl
g I decureasing

3 highs htt o Series converges via AST

b EE t isolate
FE SEE C if
I an or 2
In
n
fun 2 2

m2
2h Nh Z t n 1h27 0 for n 100 an is increasing
NZ If 0

Cannot use ASTon the Problem test is inconclusive

when AST inconclusive
fry test for div ratio test or root

test
2 In 2 0 In 2 0
Iliff in Ep
tiffs C Dn
n
21 D NE
n
James B W o x Limits Must be unique

c EE th't en
 I An In n even
Hally always positive
22
nz
2 n 4 Inch m
n Zn Inn po for nz3
Eninna N4
MY

3h17
hff En
him him o
In
The series converges via AST

Definition A series AK is called absolutely convergen

if
7 IAH fail t laal t la I t Converges If a series is
regularly convergent
Absolutely convergent it is also

Definition If a series converges but not absolutely convergent

then we say it is conditionally convergent
Show 2 things I O G series converges
z Abs Val seriesdiverges
ex IC D I CDk Ink L UK for 7 21
IC2 I 2K 3K I 3K 1

EZ Do the following Series Converge absolutely conditionally

Or diverge Stateany test used
A EE 41 Cannot userootlratio ED

EE Eti EE e series
corn
EE
I
 Gnu Viathedirectcomp test
Sincethe Abs value series converges the original series converges
Absolutely

b E c it u EE K't III f E FEEL

Ly Iz divergent e
y g2 LyEe series

Diverges via thedirectcomp test

AST I an Intl o
8h2 4

2 dd Ibn 8 32n 16h o

nffmt.la
8h2 4 L t

An is decreasing V
3 Iim 2Mt1 Bottomheavy o
ns 8h2

Original Series Converges via AST Theoriginal series is

conditionally convergent
I I EE
E orivergent e series
L EE z'm I Kc
2 the series

The original series is absolutely convergent

atb b b
Remember X Xa Xb Xa Xa or
 htt htt n 2ntz

nt21L2ntDGn7
nliffatn nliszao nliIhsnra
ijniI.Va I

lim nSn I
n x

RatioTest Let FE An be a series let n Iff f aan L

If L 4 the series converges absolutely
If L I the series diverges Cine x
If 4 1 the test is inconclusive use a diff test
Note L won't be negative b k of the abs value signs
when to use ratio test
factorials K
exponentials 2 or
C3 but not C 1
If all you have are K to number powers you will get L l

El Do thefollowing series converge diverge why State

any
tests used
A EE gun dnt I
An
n znz
I
It zntz cnty.sn
9

lion anti lim l

mo an n so g zn.cm lim q up
ns
I 3
3 n 3h
lim n the series converges via the ratio test
n sa LI
b GK An 2n Anti nth Cents
K2Gaya NZC4 n htt 42Mt1 f 4 f 4 n
n D2f 4

Anti Gentz 2ntDGD Iim Gentz 2ntDGn

2 2NH 4 n n A
L 4 n2 znH L 4 n f 4

Gns
n2fyjn

im Gritz 2ntDG thing Cinta C2ntD m

n2t2ntDL4M 4
zn y n't Intl
absual

figs stop heavy D 1 Theseries diverges via

Inn ratio test

Root Test Let An be a series let n Iff tant L

If L 4 the series converges absolutely
If L I the series diverges Cine x
If 4 1 the test is inconclusive use a diff test
When to use
7 tote powers or
III
F 2 Do thefollowing series converge diverge why State
any
tests used
aE CHIEF nliznfE hi.EE t
same lowers
ly The series converges via the root test
 C 071
4 Ettyyen diverges via ratio test

PowerSeries Aseries w the variable X A power series about

X O ta ri A
is a series of the form Egan Ao ta n t ane
for each value of
power series is a family of series one
Power series about c EE an G 4
terms t L constants X variable
Forwhat values of X will a powerseriesconverge
How to find conu P series
Ratio test C Dn
x
Root Test n LI
9 g
Geometricseries ifapplicable
a
X 9

El For whatvalues of X will the following Powerseries merge

a
EE EE an I I anti Z V X xn
ni
EntDn
hi Ha this infinite n n'it
Iim
n.net xlIodhoasuYnt.hin9 nligsEnlx
x nlinfon 1 10 0 a
1 Series will converge
no matter what value of X is used
For all X X C fans
 b An D fly int
EE t's am
GII GING
n

EM n
n'insta IL m
ntI HnL

lim n Ix Il Ix il nFa Ix H i
n y
Ntl

IX 114 for conv 0 ex 2

a
It willdiverge for Ix Il I
whatabout when 1 11 1 x o x D

EE'T EE
Don'tinclude
K I I E aiseaEs
zero
2
EE
4 41 EE
c
If EE t'M Ast

l bn In so I
converges
2 In t n 220 bn isdecreasing
3 diffIn Or Int Notation Xe GT

Def Theset of all X values that a Power series converges for is called
the interval of convergence Ioc

Def Half thelength of the Ioc is called the radius of convergence

R
A power series about X L behaves in one of 3 ways
1 converges for all X values Ioc Gro nd Rea
2 Convergesonly for X C Io C X R O
3 Converges for an interval Io C la b La b or Ca b
A C R b Ct R R BE

E 2 Find Io Ct R for the following power series

A EE kzf.IT An Anti Cnt 2 xn
nzn.tt
3
lingfant I shift ht
In f I
Thigglatff I 1 21hits 5th 1 4 t

1 12 4 1 4 c3 Xt 2 3 X al
ki 2 23 X 2 S
X t 22 3
S LX L I

E.atIaYinIsEE E I E 3ktnr
nliMflY3n DNE Diverges via Testfordivergenc

EE k
17 EE
K
37 3k n'if 3h
Is Diverges
to o

via Testfordivergenc
Ioc C s D R 1 3
Taylor Maclaurin series
El Find the Taylor series generatedby the given
function the indicated value of X a

a f x e 9 0

FG f
ex a
k We need to find f a w o f

flex f Cx
Values of K t KA dir Cx
K f
O f x e I e
l f x Ze 72 e
finding the pattern
Z F
3 F x yezxgzy.eu
get Z e
4 f4G 16 e z et

K f x Ke y 2k e
0
f Caf y f co
k
2K E
a
fo
EEof f ex ask
IE X
GE
in25
b fG 2 a L
F'G C In 2
Z
f x 1h2 Z X 2 X
 3
fill x in 2 2 sign is changing
s
power of 1h2 is inc by l

x
f'T G Cl incl Z Kth dir X
X 1 Fka C Dhana 2
1

1
I C
In 2

FGKEE.FI cx

a
EEflnuD G l
o
z
m2
I X
E
to 2K

f x a _2
K
F x fk C1
XZ Ktl
X
F x 3

fine I f G EEF ex ask

X9
K
Fut C D a
Is EE X Z

C Dk Cx 21
FI
E
K O 2kt
I