Math Real Analysis

MATH 3001 W21 A4

D ue o

n M arch

23

µ Evaluate it n t Lh ( hlxtll)

needs S da s

3ht sin4hm

Show that I , Cj!!! converges uniformly on IR but

does not converge absolutely for any XEIR .

(3) Show that neo x" converges uniformly on XE f-a, a] , for

any case but does not converge on f-1st) .

(4) Does §, , Xu-x)

"

converge on D= Co , IT ? Is the convergence

nnifrrm ?

(5) Show that the Riemann zeta formation bKk IE , thx converges

Mri firmly on [a, a) for all a>I . Show that y

' Kk - E

, buffet for x>t .

(6) Show that if IT ,

an converges absolutely, then ZI , an cos@x)

converges wmformly on R .

MATH 3001 W21 S4

ht Ln ( hlxtll) H - setfnkl-zn-sinynxg.INT converges poihtmeb

fuk 's for

Now ffncx, -31=1Nth hat 's )# tssinlnxs ) x>

-e .

3.ht AHYHX)

£ lnlnlxti ) )

+

' g ah 'm

3h 3h

E gut . lnlncxtll) 31T

+ that,

New th ( hlxtl ) )

31T = {⇒

In@Htt)) AHHH

i But

th (nlxtl ) )

fncx.it ⇐

YE , Fyi = ska .

Hence linen -H ' # HIS't .at , .÷ ,

# '"Is't 't

⇒ FIL ftp.m.ylfnkt - 431=0

Hence th-s t Mnf . on fo , IT . Thus

fly § "

fnlxldx = To 43 dx = Tho .

H By the alternating send test, the tenor converges txEIR . Moreover

,

Isn Ki - E, It!! "

I ' tent , tutti . Hence

am say, Isn in- Fie!! I --o.

N→or

Thus the series converges nntformly on IR .

Next , HER

, we have

× ! +n > In provided

na x? Hence the absolute tenets IE , ¥n diverges by

the comparison test .

X xE f-447 ⇒ HI " E ah

.

Also , ⇐ooh = Fa Carr.

So by the Weierstrass M - test , Jo xh com. mnf. as far] .

New sncxk II x" = 'II " '

, scxi-2.IM -- ÷, 1×1 Ntl (XEC-1,11)

⇒ sup Igrcxl -SCN I = snp =o.

-144 -KXSI It

- X )

so SN does not come. Wmf. on ft, l ) .

④ Ratio test for XE Cal) fixed :

• *or : M¥14 : " In "

f- i-x vase . Hence the Tents converges for X E ( o,t) .

• For X -

- o : I 0 . ( I- o) " =o

NZ I

• for x= l : 3¥, l . ( I- i )

" =o

So yes, the genets converges txt fo , I] .

New tx C- Ceil] : E ,

x ( I-x) " = x %

,

C-x) " .

So the series

converges mnifrmly exactly when the series E. ,

d-x) " does

,

for Xe [oil] . Which is the same as saying that the tents

%, y "

converges nmf. For ye foil] . We know, though, that

the latter series does not converge mnf . ( question 3)

Hence I xltx) "

does not converge Mnf . on co , if . 971

(5) fault thx = n " = e-

× Ah . For ×>a :

IfnWI E e -ah"

= Ta . But % , that converges

as it is a p -series with p

- a>1

. By the Weierstrass M test,

NE, n "

converges uniformly on XE la , d ) , ta >1 .

We want to apply the theorem about termwise differentiation . Each fncxl is differentiable, with ffCN

= ady e- 'em

= -

lenny , TX>1

. Also , Ifn (Xo) converges for some Xo ( actually, all HH

Xo> 1) . Now we check that 2£, then come. mnf . on La ,

a) . Again

by the Weierstrass M- test :

I then I s h"

= µ ha h

.

since a>1 we have a= It2e for some too . Then

Mn = 1 fun

nite he ⇐ Ite - So NE

,

Mn converges

÷ (depends onEhrtnotonn)

by the companion test ( p - tenor hath p = KE ) .

Hence I f- '

n Kl converges mnf in [ai) for all a>t.

h71

New we apply the theorem on termwise differentiation . It tells us :

Yuk NE ,

fnlx) is differentiable on Ca, a) and

94×1=33 , ff IN = -I, huh, V-xefa.no) and fast .

(6) OMG easy ! Just apply the Weierstrass M - test :

Hink Ian coslnxll E Ian I = Mn

and T Mn to as Ian is absolutely convergent .