Math Real Analysis

MATH 3001 W21 A5

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n Ap ril 1

st

(1) Let 4 : RNR be continuous , sat . 91×170 HXER and £4lNdx=1 . Show that 9h41 = nylnx ) is a Dirac sequence .

(2) Show that n!

g. fn is a Atac sequence .

>x O

'th

f

③ Suppose f d continuous on later] and f. flax"dx=o for n -

- oil, y . . . Show that full- o ttxecqej . ( Hint : use the Weierstrass approx . theorem)

(4) Find Roc and the interval of convergence for : A

zhyh N (x-4)

h

(a) I (C) I na tf na n

° xn

N

u ! (x-2) "

(b) I (d) 2- his fun)

"

n= , hw

(5) Show that ÷ , In -2 (Hint : differentiate nigh then

multiply by x )

(6) Prove that if full = oanx " is an even function

, then

an -

-o for all odd n .

H Prove that II 2 "

sin ( Tx ) converges uniformly on

Ca , a) for a >o . (Hint : recall that feign Mtf -- I .)

(8) Suppose Flxk I.oanxh, XE f-Rik ) for some R>o . Suppose that Flo ) to . Show that there are fn EIR at . GCN= fnxh

converges -

Ar x small enough , and FCNGLN = 1 .

( Remark : GIN is the send expansion of Yfcxs)

MATH 3001 W21 S5

( 1J We need to show at 9h30

es yn continuous & I. ynlxldx =L ⑦ to>o

, , ,§g9nlHdx →o as n -so . x

obviously Yuko and 9h iscouhhuom . Moreover,

I ynlxldx = Ian 41nA DX f! 4187dg =L . y=nx

So conditions c) & H) are satisfied . Next ,

µ§g4nlNdX=!!h9lhNdx + In ylnx) dx = Iandelyldy + §

,

my)dy .

A

f nf%y1dy Since the totalarea

thx * x'stamin} 'atthefreaecanes

has →a smaller and smaller as him

so clearly try §9nlNdx=o . Same tushy for I! ynlxldx . so properly ③ alto holds ' me fn are not continuous

,✓ so this is Nota Drac sequence as we defined it

µ The properties a)4121 are ohr§sG correct . Next, in clan . My *fog fnlxldx =o if n> Yg .

Mikkel. (But . . . often the continuityrequirement is dropped . . . )

(3) Since fat fix) x" DX -o th then §fCx) pcxldx ⇒ for all polynomials p. By the Weierstrass approx . theorem, too

F- p at . IHH -plxl KE txtGift . Then

o= Saerfcxipcxidx = fit"'ll "g"Itf:!!!

'

'd . .+

a

But I Sabflxllpcx)-fexildxf ± Sat lfcxye E EM (f-a)

,

where M -

-

sup H1N1 . Xtfqf]

Hence fab fix, 'd, = - Ifix) (pen-fin)dx

E E Mlf-a) .

This holds te , so fab flat 'dX=o . Suppose IX.Etat]

S-t . ffk! to , then f 2=970 . Since fell

' is continuous

f-4×1 If>o sit . Ix-also implies Titian , flxl

' > I . Hence

Xo q Koto

Sffxtdx > f Azdx >o. A

xp-J

Tris contradicts fab 'dx=o . We conclude that flxko AXE faff .

(4) (91 L= ahmfnp 2 n -4h

= 2 ⇒ 12=1/2

×= '

g : 2- 2414

"

cow. ; x=-tz CONV.

n'll hi

⇒ interval of convergence is [ -

I , I] .

Ay L= alimony (ennj ' = O ⇒ pea

⇒ tht of com is f- ga) 3 5

L = lsmfmp n - ' th = I ⇒ 12=1 envoy

Here 5- 4 , so come . for XE 13,5) 4

A- 3 : n ELI CONV; e-5 : NE

, 'T DIV

⇒ Mt . of com . is [3,5) .

oh use ratio test :

Cnts) ! Ix-4h41

*yn . mix-yn = Ix-21 Ca)

" EY 1×-21 . te

⇒ car. for Ix-21 tea , i - e . Hike

⇒ pie , C=2 , Carr. for xe (e-2, e-12)

° X= e-12 : ÷, n÷ n ! ~ KITT (F) h

Stirling's formula n ! eh In

~ filth →N as h-7N , so the rents diverges .

° x= e-2 : I n ! (e-4)

"

Ma n"✓ gain II le-4) " t hh

= HTT C- ke ) "

Hence the send conveyor ! = KITTED" (¥ - ith

w

Int - of com : (e-2, eez) . I 0.4851

this converges to zero expon .

quickly 1h n : e- n ten 0.481

(5) § .

xh com. Wmf. 1h XE [ I-a, Ha) , VIE lo, s) .

So dy, ox ! E ht

"" =

% ,

n x ""

= tx LE , nx "

(Ho ) Take x' 'k : ta d-dxq.im/*z--nE, In in = fly ¥, = ¥2 = 4 for A- 42

.

(6) f-Cx) = fl-x) ⇒ NII an x" = II.oeijhanxh ⇒ NII ft - I- ith ) anx

" =o tx

But if nztobnxh - o tx in a neighbourhood of x=o, then the th ( take x- derivatives at x-o) . Hence we have

11 - l-t ) "

) an --o th . So for n odd : 29N =o ⇒ an -o .

H 12h sin (Tx ) I = 2h In, I sin 3¥

It ⇐Y '

y l

3 " x

-

sing ITH

⇐ ⑤ "

ta t x>a

Hence the series converges uniformly on XE la , a) by the Weierstrass M test .

(8)

Flop a.to . FCNGCM = n ?÷ Cnt

"

, Cn -

- II an-efe and 6=1

, ch=0 Fn > I ( HN Glx) =L) . So fo = Yao

and bn = - at . II. ane be th> I .

Now 19kt EM Hk, so

ienkiaiiiiieei ' iii.÷, Eileen ⇐sina.in :÷÷÷÷. .

Kota . ,

⇐ ii. ⇐iii.÷ .

iv.wanna. i÷÷÷÷÷ . . a -

first him : ek-I

the next mm is : eh , g. {

"" xdx = HII

d

'

ex-a

o l l l l l l S Lk-,

T Gi '

me next mm : iii. e÷ s ta ! -'

x'da -- ÷, er. '

,

and soon .

so i÷÷÷i÷ . . . iii. s ¥, . Mas Hnk pn=a÷, IF. .)

"

th .

.

Now E. Pn lxl " so ? Radio test : ftp.hn-I/lxl → L :

" t.ia.ci::: '

ii. ÷. ' ie. ⇒ L

- - Mae, Kl . So for KK

"

j! tee O

series converges !