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Elementary analysis

Elementary analysis
You may work in teams of size no more than three. Each team should hand in
a single homework set, signed by all of the team’s members. All team members will
receive the same score for this homework. The value of this homework set is 15 points.
(**) Unless otherwise directed, you may use results proved in class notes or in Rudin,
but if you do, cite them as [Rudin, Theorem 5.7] or [Class Notes, Theorem 8.3].
(***) Write up your solutions in mathematical English (as in our class notes) and in
publication-ready form, preferably typed in some technical word processing language.
1) The comparison test
(which we will prove in class) says if 0
a
n
b
n
for suciently large
n
and if
1
1
b
n
converges, then so does
1
1
a
n
, and furthermore if
1
1
a
n
diverges, then so does
1
1
b
n
.”
Show how to use this test to compare the series
1
2
1
n
2
n
+1
and
1
2
1
n
1
:
5
. This will require you to
prove an inequality.
2) In this problem we will use the comparison test from problem (1) to prove the Limit Comparison Test
which says: Suppose
a
n
>
0 and
b
n
>
0 for all
n
. If lim
n
!1
a
n
b
n
is a positive real number
L
, then
either both
1
1
a
n
and
1
1
b
n
converge, or both series diverge.” Here are the steps:
2-a) Given that
a
n
>
0 and
b
n
>
0 and lim
n
!1
a
n
b
n
is a positive real number
L
, explain how we know
that for suciently large
n
, we have 0
< 0 : 5 L < a n b n < 1 : 5 L . 2-b) Use 2-a) and a comparison test to show that if 1 1 a n converges, then so does 1 1 b n . 2-c) Use 2-a) and a comparison test to show that if 1 1 a n diverges, then so does 1 1 b n 2-d) Use the limit comparison test to show that 1 2 n n 3 1 converges. This will require you to specify a series whose convergence or divergence you already know, and then show that the limit comparison test applies, and nally make a conclusion about 1 2 n n 3 1 . 3) The Integral Test (which we will prove in class) says that if f ( x ) is a decreasing non-negative continuous function whose domain includes [1 ; 1 ), then 1 1 f ( n ) converges if and only if lim T !1 Z T 1 f ( x ) dx < 1 : [Warning: lim T !1 R T 1 f ( x ) dx is sometimes written as R 1 1 f ( x ) dx but it must be evaluated using limits rather than trying to use arithmetic with innities.] Use the integral test to determine convergence or divergence of each of the following. You may assume that the functions involved are continuous. 3-a) 1 1 1 n 1 : 5 3-b) 1 2 1 n log( n ) where log( n ) denotes the natural logarithm of n . [Hint: try u = log( x ).] 3-c) 1 2 1 n (log( n )) 2   56 nursing care plans To present using one ofthe nursing care plans from assignment 1 and refer to the ANMC domains in order to identify why and how nurses need to be cultural sensitive when caring for this patient who you can decide could t non-English speaking background (ARABIC). (10 minutes) Refer to the ANMC domains in order to identify and discuss why and how nurses need to practice in relationship to legal and lor ethical issues and to refer to either a health topic or an incident relevant to either a maternity (eg l/F) or paediatric eventls (eg medication error). (10 minutes) The tutorial presentation is to be interactive with the audience but based on sound knowledge and research based evidence and demonstrate correct APA format for in- text and end-text referencing Evidence of research, using a variety of current credible resources This will be a total of 20 mins, a power point type slide show is acceptable however the presentation may be as creative as you like. A lesson plan and copy of any power points or any handouts is required to be presented to the tutor on the day of your presentation Review the marking criteria to achieve maximum marks for this assessment. ORDER THIS ESSAY HERE NOW AND GET A DISCOUNT !!!  

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