The geometric Series

The square of side length I has area I .112

" 4 It can be covered by infinitely many46 % triangles of areasIz , 14 , Ig , IT , - - .132

As a result the equation"

Area of the square"

=

"

sum of areas of the triangles"

een be written as :

I = It Iz t Is t ¥ t - - - = §,

(E)"

.

as

he series on the r h s is of the type &xh . Suchk= 1

series take the name of geometric series of ratio x

↳o

By elementary algebra it holds that for x # i

1 t X t X'

t . . . . t I= 1 - x" "

#( prove it by induction )

nwt IAs a result I xh = - I andhe = i r - Xh htthm E x

"

=ein . a = ly

to × "

h → tox he =L h → t -h - ×

Ix - t o excl

Ln the case x= a 27,1"

= to trivially .

The previous computation shows that the geometricseries converges for XE Con ) ( in the example of the

covering of the square x=lq ) ,while it diverges here X > l .

The following results , known as convergence anterior ,are obtained comparing a series

with the geometric series . .

Proposition ( D'

Alembert criterion )

let Za be a series with a 30 . tf there eaizts o - g atk K

such that

dares g asymptotically ,

k

then Zakconverges .Proposition ( n - th root criterion )Let Ianbe aseries with a#o . Ff there exists getsuch thatfars g asymptotically ,Kthen

Zakconverges .

Before discussing Thun , we state and provea necessary condition for the convergence of

a series

Proposition ( necessary condition for convergence )let

Eatbe a series with

an30

. If

Ean convergesthen him a,

= 0.

K

Pref : Suppose by contradiction that lineman = a > o .

By the definition of limit with E = age ,F N @lNs.t .

for all k 3N - agg an - a c age . In particular an > age-

tf k > , N . This says that

se an = IE ant Ewan > Ct E In - N ) =w

- X

" asC

= C - azn taz . w

linin Sir 3 C - I w it E . linin w = to contradicting the

assumption that Z' an < to

Example i the series Itu,

¥2 diverges since

aw =I -0 I fo

.

utz w - s to

Remark : The necessary condition is not sufficient .I

Consider the series I In ( harmonic series )k=r

Thesequence an = In → o , but 21h = t !

To see that,

consider thefollowing argument ia

I

feltI t ⇐+ 141t ⇐t f- tf tf) t - - - 3k=1

i t It ¥t 14 ) -1 ⇐tf t tf tf) t - - . tf; tf ; t - . - - the; ) t - - --

zit week

In the previous sum we recognize that the term 42 appears

infinitely many times , sothot the series diverges .

Proof of D'

Alembert criterion

Besides finitely may terms , that do not change the asymptoticbehaviour of a series , we can assume that agent E g .

Then are99 , as E 992 E glean , . . . an E g ".'

a,

• A A

he

⇒ I'

an e an. E g

" - '= a

,- E q a t no

h = I 4=1 h = o-

geometric series with g - I

D8

Example Let x E Coin ) . Study the asymptotic behaviour of

It,

n x

"

. By D'

Alembert with are u x"

we have that

Waze = =x - ( ¥ )

Ea I ⇒ Enact no

.

( ¥ e II ⇒ Ee ×¥ - a = IF ⇒ n > Fe)

Telescopic Sues

Consider IF,

÷u , = t.ztf.z-tf.at - - -

Noticethat ÷j = ÷ - ÷,

.Then

E. ' ⇒ = Eit.⇒ = a ¥¥¥¥¥¥¥ . .

In general sea ( I - ⇒ = e - ItIfIft. . . ÷ ,= 1 - 1-

htt

⇒ E;a¥= him sue lain ( i - ÷ ) - - s .

Using the telescopic series one een show that

I:÷ . . . a at E ' , III." 'E.IE#a)Thus su - If

,

wt e s 'n=§q÷u, I 2

By the comparison him Su E 2 . It een be proven thot

I. at - I .+ no

atMore precisely it holds that IF Tex { a + • a > aThis series is knows as generalized harmonic series -

Alternating Series

Let an> o the IN . Consider the series

⇐affair .= - ant as - as tay - - - -The following Leibniz Criterion holds true :

Proposition : Assume Can) decreasing ( ant , Ean ) andusuch that binman - - 0 . Then E C- Jean converges .

W =/

Example : If the "tz= A - It } - Iet - -Converges since an = In is decreasing and binman = him In = 0 .

Functionsof one variable

We consider X ER,

Y ER and we consider a

mop f : X - o Y such that to every a E X

( the set X is known as domain of f) it associates only oneelement f Cx ) E Y ( the set Y is known as co - domain

off ) .fXo x f Cx ) E Y

f Cx ) is also said to be the image of x through f .

The set f = L f Cx ) , xe X } is the set of all theimagesthrough f of points x e

X.

Note that in general f ( X ) F Y .

Ft A F xz =D fkn ) f fcxz ) t is injective in X

If Fye Y F x E X such thot y = FG ) , then

f is Surjective on Y

If f is both injective and surjective , then f isa bijection or one - to - one mop .

Gf= L Cx , y ) e R

'

: x EX, y = f Cx ) } E Xx Y is the

graph of the function f . It is a subset of R2 .^

ayGraph of a function This is Not

the graphC x , fix , ) of a function

f¥•TMaginot 'sS >x × x then one

image -

Examples

f : 112 - o R : R a x i > f Cx ) = XZ E 112y l

is not injective ( - t t 1 but f C- a) = fu ) = I )I 1is not surjective on R ( - 2e

"

Ed,

but Axe IR : XE - 2)^

a

Fa Cx ) fzlx )

y =

a.twoinfections y ! gY.fm/Yn!?m!ssechioh- l tr X parallel to the

x axis

y =- 2

>

no intersection x

f : [ o , tho ) - o To , too ) : [ o , too ) a x I - a f Cx ) = x2 e Entre )2 2

is injective in [ o , too ) and sur je dive on Eo , ere ) .Such a function is one - to - one -

Inverse function

A function f : X - o Y one - to - one on Y is also said

to be invertible . The inverse function f-h

is then

defined as : ft : Y - o X the map that associatesto a given ye Y the unique xe X such thot y = f Cx ) .

Ftheds

f C f'

ca ) = fifth ) ) = ×× #f ×

Eplex f-

'Cx ) # x

f i [ o , to ) - o To , to ) x c- E. to ) - o fcx ) = the G. to )

is one - to - one on [ o , t no ) . We now look for its inverse .

f' '

: fate ) -0 Ei too ) :

ye Geno ) - o f- Ly ) = X e Eo , to ) : y=x2Since y is fixed we need to Solve y = XZ for XE E. to ) .

We have Shot x = tf , so that

f-'

G) = TyWe plot the graph of f

"

using asits argument

the variable x, namely f-

'Cx ) =D

§ . , = Lay ) ER? xe Co , to )

, y = F)^

a ÷× ,× rx-t.ae Ex

× t→×2 if =xi

I 2

x x

Limits of functions

we now define the meaning of Iggy Fox ) =L ,assuming that f i X → Y and × . is such that

V-8> o ( xo - S , x. t s ) I Ko } n X ¥ § .

HE > o F 8=8 so : tx E ( Xo - S , Xo t f) I 1×03 n X

If G) - Llc E ( i.e . L - es f Cx ) c Lte ) -

y^

y = Lee Iy =L - E III• I • >

- E Xo Xo -1 E X

Wesay that a function f : X -0 Y is continuous

at a point x. c- X # Lingo f Cx ) = fcxo ) .