Nov.

3 / 2017-

a

Ex ( From quiz )

an = tan ( 2in . 17 . +fg ) .

lim an = ?n → a

data'IIiIII÷s?

?

isno±±a⇒book at f ( × ) =

tan ( 2.gg# ) .

xlnjmaton (2@÷×) = tan ( ¥ ) ( continuity

Listen= * = ¥= '

.aEn±÷. Baoktoseniesconnergencetests

•

Alternatingseriesb, , bz , bz ,

- - - . bn 30

b,

- bz + b3 - by + bg . - - . - ?

E± bn =nt ( alt .

harmonic

I - ±+ } - 4- + . . .

corn.

series )

( harmonic

1++2+3 + . . - .

div. series )

-

Notation oo hit

b,

- bztb } - by + . . = [ f) bnn=l

C- 1 )"

-1 , +1,- 1

g^ - -

- .

nttna +1 , -1 , + 1 g- 1

, . --

H ) =tAH.seriestT.msleibintzb

, .bz/b3i - - -

�1� bn > ofor all n .

�2� b, 2 bzzbz - - - decreasing

�3� lim bn =D

n→

:Then [ C- IT "b,

=b,

- bztb }- by + . - .

.

n=1

is convergent @

¥ bn =h- D 1/+2 , } . - - > 0

�2� decreasing

�3� ntgmo'T =o

I - tztzt - ¥+ . - - converges(b) atttessefies) .

( Infact, we will see later using Taylor series

I - fist - It - - - = ln (2) ) .=mlificationofaltseniestest

#qblb

,

#b

,- bz

bz _•b3 b,

- bztb }bq•s- b,

- bztbz . by

-3g:

: :

:

If limit of alt . series !

-n

E*±§gH,2÷n- bn = End

lim -21 =- 2 1=0

n → a Itn

Se alt .series test does not

apply .

-

a hit�1� n3n÷ > 0

EI [Hnn3÷l@ lim nn÷t=°

n= I n→a

�3� dec . ?

f ( x ) =××z÷ - f{ × ) = 2×471-543×2=(113+1)2

=×(2-X] - × > 2 then flex )< 0 .

⇐3+42

( It is ok if decreasing starting from n=3 , . . )

-Absolwteconnengenucofaseriesja

, 1921 - - - an positive on negative .

Def § An is" absolutely convergent

"

if §|an|is convergent .

Ex= §4)"

+4 is convergent

but note absolutely convergent .

( §'T is div . ) .

. §an is"

conditionally Conn .

"

if

it is conu .but Lotabs

. Conn.

÷every If {lanl is conv

.then

§ an is Conn .

Or if abs.

Corn . => Conn . ( in usual Sense ) .

.

Next time : Ratio test .

TelRatio test ) ,

§ an

¥;z1an¥l=L< 1 then fanautos