Curriculum Development W7 Learning Objective Examine a lesson plan according to the required content and ideas associated with standards and bloom's taxonomy. *Digital Dropbox - Assignment Develop a plan for how you would teach one of the instructional methods on page 2 of the Voltz textbook in your classroom environment. *Discussion Board Discuss and demonstrate how you would apply the method in the classroom environment. MATH 3001 W21 A1 Due o n Fe brua ry 3 rd (1) Let {xn } be a real sequence . Show the following : {Xn) is unbounded ⇐ there is a subsequence {Xn , } at . lim Ku ,K-so I =D (2) Find tho real sequences Exist, {un ) such that xn →x, yn →y and xn s yn hit x>y . (3) Find the tail of the gents II. am, where a E C-bi ) . That d, calculate exphvitty a En = I at . f-htt (4) Suppose a sequence { xn) has two hitsquinces lack} and {bet such that every xn is either an 9k or a fee . Suppose that iahjmak = a and effy be =p . show that the sets of limit pouts of Hn) equals S= {a, pig . (5) Let Am = FIX " , XE tht ) . Prove that {Amy is a Cauchy sequence . MATH 3001 W21 S1 H ⇐) Ik sit . HH >I (because {Xn } is unbounded)I 2 Then among all indices k>he, Eh, at . 14,132 ( if that was wet the case , then fling would be founded !) . Then annoy all k>kz I kg sit . Hk, 133 . ETC. So : HAEIN E kn Sot . lxknl > ' n . Also , Keck! kzs - - ' . Thus fkn ,4, is a subsequence of fxnlgn with him Hn,d=N K ⇐) TM>o E R sit . Kyl>M (as Ha! -9%) . Hence {xn4 cannot be bounded . (2) Xn= - th , yn = th , X=y=o . ③ Am = FI,at = It at . - +am = I - am " i -q - A- = ¥n,pAm= Fa = II. ai En = A - An = 9h41 I - q ' ④ By definition of the sets, we have {a, pig c S . Now we show that { a, pips , namely that any LES must be equal to a or p . Let xn , be a convergent subsequence of xn , with fruit L . Each of the xn , is some as or some be Ifor some index s,t). • suppose infinitely many of the xn, (as K varies) are from both He bit of the as and the bee . Then both the 9k and be have to converge to L, so a=p=L . ° Suppose the xn, contains only Amlely many of the ap . Hence it contains infinitely many of the be . So L=p o If Xnk conform Amlely many of the be only , then a=L . (5) I Am-An 1=17×1 Iam"-x "" I take m>n WoLOG = III / km -n - it E 1¥ , lxlh " "→ ° yo . MATH 3001 W21 FINAL EXAM : 4) Find all a >o such that the following tones converges : C-opt] z lnfn " ) NII qh 4) Suppose that the series Ian converges conditionally and h? I A-opt] set pn= ; ( lannan ) . Show that Inpn diverges . (3) For a>o, decide if the following genes is absolutely convergent, convergent or divergent : G-opt] n C- yh ( lnnjlhh ya - (4) Let fnlxj = n fn It'm ' hx , ✗>0 . Ppt] (a) Find the pointwise limit function f-1×1 . Apt] f) Does fn converge to f uniformly on lo , a) ? (5) Consider 1--1×1=2%1×1 4-pt) (a) Find all ✗ER so that the series converges . [b-pt) (b) Sh ...

1. Curriculum Development W7
Learning Objective
Examine a lesson plan according to the required content and
ideas associated with standards and bloom's taxonomy.
*Digital Dropbox - Assignment
Develop a plan for how you would teach one of the instructional
methods on page 2 of the Voltz textbook in your classroom
environment.
*Discussion Board
Discuss and demonstrate how you would apply the method in
the classroom environment.
MATH 3001 W21
A1 Due o
n Fe
brua
ry 3
rd
(1) Let {xn } be a real sequence .
Show the following :
{Xn) is unbounded ⇐ there is a subsequence {Xn
,
}

2. at . lim Ku
,K-so
I =D
(2) Find tho real sequences Exist, {un ) such that xn →x, yn →y
and xn s yn hit x>y .
(3) Find the tail of the gents II. am, where a E C-bi ) . That d,
calculate exphvitty a
En
= I at
.
f-htt
(4) Suppose a sequence { xn) has two hitsquinces lack} and
{bet such that every xn is either an 9k or a fee . Suppose
that iahjmak = a and effy be =p . show that the sets
of limit pouts of Hn) equals S= {a, pig .
(5) Let Am = FIX " , XE tht ) . Prove that {Amy is
a Cauchy sequence .
MATH 3001 W21
S1
H ⇐) Ik sit . HH >I (because {Xn } is unbounded)I 2
Then among all indices k>he, Eh, at
. 14,132 ( if

3. that was wet the case
,
then fling would be founded !) . Then
annoy all k>kz I kg sit . Hk, 133 . ETC. So : HAEIN
E kn Sot . lxknl >
'
n . Also
, Keck! kzs
- - '
. Thus fkn
,4,
is a subsequence of fxnlgn with him Hn,d=N
K
⇐) TM>o E R sit . Kyl>M (as Ha! -9%) . Hence
{xn4 cannot be bounded .
(2) Xn= - th
,
yn
= th
, X=y=o .
③Am = FI,at = It at . - +am =
I - am
"
i

4. -q
-
A- = ¥n,pAm= Fa
= II. ai
En
= A - An
=
9h41
I
-
q
'
④By definition of the sets, we have {a, pig c S . Now we show
that
{ a, pips
,
namely that any LES must be equal to a or p .
Let xn
,
be a convergent subsequence of xn , with fruit L .
Each of the xn
,
is some as or some be Ifor some index s,t).

5. • suppose infinitely many of the xn, (as K varies)
are from both
He bit of the as and the bee . Then both the 9k and be
have to converge to L,
so a=p=L .
° Suppose the xn, contains only Amlely many of the ap . Hence
it
contains infinitely many of the be . So L=p
o If Xnk conform Amlely many of the be only , then a=L .
(5) I Am-An 1=17×1 Iam"-x
""
I take m>n WoLOG
= III / km
-n
- it
E 1¥ , lxlh
" "→
°
yo
.
MATH 3001 W21
FINAL EXAM

6. :
4) Find all a >o such that the following tones converges :
C-opt]
z
lnfn
"
)
NII qh
4) Suppose that the series Ian converges conditionally and
h? I
A-opt] set pn= ; ( lannan ) . Show that Inpn diverges .
(3) For a>o, decide if the following genes is absolutely
convergent,
convergent or divergent :
G-opt]
n
C- yh
( lnnjlhh
ya
-

7. (4) Let fnlxj = n fn
It'm
'
hx ,
✗>0
.
Ppt] (a) Find the pointwise limit function f-1×1 .
Apt] f) Does fn converge to f uniformly on lo , a) ?
(5) Consider 1--1×1=2%1×1
4-pt) (a) Find all ✗ER so that the series converges .
[b-pt) (b) Show that FIN is continuous for ✗ c- C-1,1 ) .
(6) Use the Dirichlet test to show that Fogh
é
"
✓ht✗T
Copt converges uniformly on [90) .
(7) Find the Taylor series of f-1×1=
h×
✗
I
✗ 20
,
about
the point c-1 .
["Pt)

8. what is the radius of convergence ?
Hotel : 70 points)
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HW9
Solution
- Assignment solutions
Real Analysis II (Memorial University of Newfoundland)
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HW9