Homeworking

H Zn -- (357-1"

a) 3Fzi= 2¥ . ht÷ = Es+

Eg

=

EseiZ

,=

I.ei #

3•

•

ZE

-49k¥•

⇒ =.

ees¥ @÷÷#:

b) By

partial, zn=( e" . Since ei is on the

unit circle for all n and I

k¥3 Zn = O .

3

3.)let Su be the wth partial sum of E. 2n .then uh;zSn= S = § ZnMoreover

,the Nih partial sum of ⇐ En is

I , t Izt - - - t In = Zf---tZI = IN

thus k¥5 = 5 and so E. En -- 5

4.)a) In E converges when Izkl and divergeswhen I # ZIWhen AKI , EEE = IzTrees E.E' =/Izzy- I = Iz - I = Fzn =b) let z - - reio , o - rat . Since EEE is convergent,€, E - - ⇐ Meine = ⇐ ( rncoscnottirnsincnot )= ⇐ Mastro ) ti In Masino )c) I , Masino ) ti E. Moshe ) = E. E = Eez= EE . = Ez ' EEE = Iifa=

=i

→ ⇐ rncoscno ) = & ⇐,rnsmcnoj-rs.in#I-2rcoso-tr2

I - Zrcosftr

5.)(a) Notice , E. ( TTY is a geometric series .

Since I Fit = HIT = Zz > I , ⇐ I E.) n divergesAlternatively , the divergence test can be used .

(b) ⇐ I I = E. he converges by the Calc # p - series testx Egni

thus € eye is absolutely convergent andEni

thus em is convergent .

(c) We'll use the Dirichlet test .let an = ht and bn= e'

Eni

then ant , can for all n and Info an -- O .

Consider the Nth partial sums of InbnBe = e' EI i

,BE Eti t e'

"= i - I

,Bage' Ei + e' ite⇐ i - I - i = - I

By = e it e " it e ite " ' = i - I - it 1=0

Bs - = i,

BE i - I , Bz = - I , Bs = O , - - . - - -

Hours Bn is either i,

i - I,

-1,

or O for all N .⇒ I But =/ Ibn ) E Ii - it -52 for all N

n - - I

thus by the Direhlet Test , EI is convergent .

(d) Since kiss- I

2- Bti ) n'

= # O,

€.

iI7 diverges by the Divergence Test .

6.)n€ in Get 2in

a) keys" fizzy , = him

" =k±sHtzI=hzil

By the root test , the series convergesif 12¥24 c I ⇒ Iztzil c 2

Tries the radius of convergence is 2and the disk of convergence is 12-+2-122

I • - Zi ,

- 4111\

-

"

b) On the boundary of the disk of convergence ,12-+24=2

.

thus Eng I iEn f- kiss ' ZE = 4=1--1*0Hms keg it

"

# O, by problem 2 .

⇒ ⇐ in Cztzil"

diverges when 2- is onthe boundary , by thedivergence test .

A

A

Eoe¥nEn-aik.nl#EiiEI--E.mEH--tsH( nth

By the ratio test , the series converges if ztlzk I⇒ Hk 3

Tues the radiusof convergence is 3

and -

thegeigy,of emergence

b) let Z be on the boundary, so I #=3 and suppose z ¥3 .

Let an -_ In and bn . - Z=fzYthen anti > an and Keath =D .

Are Nth partial sum of €,

bn is

Bu-_¥zt¥7 - - it (E) N--FEET# ' t - - it LET ) (E¥4=⇒ IBut =/

"

I sl*¥

=-2 fraudIt- ZI It - Est

thus, by the Dirichlet Test , the series

converges .

If z =3,

then £ cn¥E=£o¥ ,which diverges by the Calc I p - series test .

8.1

Initat I f- ⇐ II, # = #F

By the riatio test , the series converges if I # 4and diverges if I # 71 .

Thus the radius af convergence isI

and the disk of convergence is IZKI

b) On IZH , I EY --

Fantwhich

I 1

converges by the p - scenes -6A .A

⇒ E innit is absolutely convergentn - - op

and thus convergent .