009C Sample Midterm 3 - Revision history

MathAdmin at 16:40, 28 November 2017

2017-11-28T16:40:46Z

MathAdmin at 18:39, 28 July 2015

2015-07-28T18:39:05Z

MathAdmin at 21:03, 28 April 2015

2015-04-28T21:03:49Z

MathAdmin at 22:25, 26 April 2015

2015-04-26T22:25:45Z

MathAdmin: Created page with "'''This is a department sample midterm, and is meant to represent the material usually covered in Math 9C through the midterm. Click on the
2015-04-26T18:48:15Z

Created page with "'''This is a department sample midterm, and is meant to represent the material usually covered in Math 9C through the midterm. Click on the <span class="biglink" style="color:..."

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'''This is a department sample midterm, and is meant to represent the material usually covered in Math 9C through the midterm. Click on the  boxed problem numbers  to go to a solution.'''

'''In-class Instructions:''' This exam has a total of 60 points. You have 50 minutes. You must show all your work to receive full credit You may use any
result done in class. The points attached to each problem are indicated beside the problem.You are not allowed books, notes, or calculators.
Answers should be written as <math style="vertical-align: -5%">\sqrt{2}</math> as opposed to <math style="vertical-align: -3.5%">1.4142135\ldots</math>

== Convergence and Limits of a Sequence ==

[[009C_Sample_Midterm_3,_Problem_1| Problem 1. ]]  (12 points) Test if the following sequence <math style="vertical-align: -10%">{a_n}</math> converges or diverges. If it converges, also find the limit of the sequence.

::<math>a_{n}=\left(\frac{n-7}{n}\right)^{1/n}.</math>

== Sum of a Series ==

[[009C_Sample_Midterm_3,_Problem_2| Problem 2. ]]  For each the following series find the sum, if it converges. If
you think it diverges, explain why.

::(a) (6 points)      <math style="vertical-align: -50%">\frac{1}{2}-\frac{1}{2\cdot3}+\frac{1}{2\cdot3^{2}}-\frac{1}{2\cdot3^{3}}+\frac{1}{2\cdot3^{4}}-\frac{1}{2\cdot3^{5}}+\cdots .</math>
 

::(b) (6 points)      <math style="vertical-align: -75%"> \sum_{n=1}^{\infty}\,\frac{3}{(2n-1)(2n+1)}.</math>

== Convergence Tests for Series I ==

[[009C_Sample_Midterm_3,_Problem_3| Problem 3. ]]  Test if each the following series converges or diverges. Give reasons
and clearly state if you are using any standard test.

::(a) (6 points)      <math>{\displaystyle \sum_{n=1}^{\infty}}\,\frac{n!}{(3n+1)!}.</math>
 

::(b) (6 points)      <math>{\displaystyle \sum_{n=2}^{\infty}}\,\frac{\sqrt{n}}{n^{2}-3}.</math>

== Convergence Tests for Series II ==
[[009C_Sample_Midterm_3,_Problem_4| Problem 4. ]]  Test the series for convergence or divergence.

::(a) (6 points)      <math>{\displaystyle \sum_{n=1}^{\infty}}\,(-1)^{n}\sin\frac{\pi}{n}.</math>

::(b) (6 points)      <math>{\displaystyle \sum_{n=1}^{\infty}}\,(-1)^{n}\cos\frac{\pi}{n}.</math>

== Radius and Interval of Convergence ==

[[009C_Sample_Midterm_3,_Problem_5| Problem 5. ]]  Find the radius of convergence and the interval of convergence
of the series.

::(a) (6 points)      <math>{\displaystyle \sum_{n=0}^{\infty}}\frac{(-1)^{n}x^{n}}{n+1}.</math>

::(b) (6 points)      <math>{\displaystyle \sum_{n=0}^{\infty}}\frac{(x+1)^{n}}{n^{2}}.</math>

@@ Line 1: / Line 1: @@
-'''This is a department sample midterm, and is meant to represent the material usually covered in Math 9C through the midterm.'''<br>'''Click on the <span class="biglink" style="color:darkblue;">&nbsp;boxed problem numbers&nbsp;</span> to go to a solution.  An actual test may or may not be similar.'''
+'''This is a sample, and is meant to represent the material usually covered in Math 9C for the midterm. An actual test may or may not be similar.'''
-'''In-class Instructions:''' This exam has a total of 60 points. You have 50 minutes. You must show all your work to receive full credit You may use any
+'''Click on the''' '''<span class="biglink" style="color:darkblue;">&nbsp;boxed problem numbers&nbsp;</span> to go to a solution.'''
-result done in class. The points attached to each problem are indicated beside the problem.You are not allowed books, notes, or calculators.
+<div class="noautonum">__TOC__</div>
-== Convergence and Limits of a Sequence ==
+<span class="exam">If it converges, also find the limit of the sequence.
-<span class="exam">[[009C_Sample_Midterm_3,_Problem_1|<span class="biglink">&nbsp;Problem 1.&nbsp;</span>]] &nbsp;(12 points) Test if the following sequence <math  style="vertical-align: -10%">{a_n}</math> converges or diverges. If it converges, also find the limit of the sequence.
+::<math>a_{n}=\left(\frac{n-7}{n}\right)^{\frac{1}{n}}</math>
-::<math>a_{n}=\left(\frac{n-7}{n}\right)^{1/n}.</math>
+== [[009C_Sample Midterm 3,_Problem_2|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 2&nbsp;</span>]] ==
-== Sum of a Series ==
+<span class="exam">If you think it diverges, explain why.
-<span class="exam">[[009C_Sample_Midterm_3,_Problem_2|<span class="biglink">&nbsp;Problem 2.&nbsp;</span>]] &nbsp;For each the following series find the sum, if it converges. If
+<span class="exam">(a) &nbsp;<math style="vertical-align: -50%">\frac{1}{2}-\frac{1}{2\cdot3}+\frac{1}{2\cdot3^{2}}-\frac{1}{2\cdot3^{3}}+\frac{1}{2\cdot3^{4}}-\frac{1}{2\cdot3^{5}}+\cdots </math>
 <br>
-::<span class="exam">(b) (6 points) &nbsp;&nbsp;&nbsp;&nbsp; <math style="vertical-align: -75%"> \sum_{n=1}^{\infty}\,\frac{3}{(2n-1)(2n+1)}.</math>
+<span class="exam">(b) &nbsp;<math style="vertical-align: -75%"> \sum_{n=1}^{\infty}\,\frac{3}{(2n-1)(2n+1)}</math>
-== Convergence Tests for Series I ==
+== [[009C_Sample Midterm 3,_Problem_3|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 3&nbsp;</span>]] ==
-<span class="exam">[[009C_Sample_Midterm_3,_Problem_3|<span class="biglink">&nbsp;Problem 3.&nbsp;</span>]] &nbsp;Test if each the following series converges or diverges. Give reasons
+<span class="exam">Give reasons and clearly state if you are using any standard test.
-::<span class="exam">(a) (6 points) &nbsp;&nbsp;&nbsp;&nbsp; <math>{\displaystyle \sum_{n=1}^{\infty}}\,\frac{n!}{(3n+1)!}.</math>
+<span class="exam">(a) &nbsp;<math>{\displaystyle \sum_{n=1}^{\infty}}\,\frac{n!}{(3n+1)!}</math>
 <br>
-::<span class="exam">(b) (6 points) &nbsp;&nbsp;&nbsp;&nbsp; <math>{\displaystyle \sum_{n=2}^{\infty}}\,\frac{\sqrt{n}}{n^{2}-3}.</math>
+<span class="exam">(b) &nbsp;<math>{\displaystyle \sum_{n=2}^{\infty}}\,\frac{\sqrt{n}}{n^{2}-3}</math>
-::<span class="exam">(a) (6 points) &nbsp;&nbsp;&nbsp;&nbsp; <math>{\displaystyle \sum_{n=1}^{\infty}}\,(-1)^{n}\sin\frac{\pi}{n}.</math>
+== [[009C_Sample Midterm 3,_Problem_4|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 4&nbsp;</span>]] ==
-::<span class="exam">(b) (6 points) &nbsp;&nbsp;&nbsp;&nbsp; <math>{\displaystyle \sum_{n=1}^{\infty}}\,(-1)^{n}\cos\frac{\pi}{n}.</math>
+<span class="exam">(a) &nbsp;<math>{\displaystyle \sum_{n=1}^{\infty}}\,(-1)^{n}\sin\frac{\pi}{n}</math>
-== Radius and Interval of Convergence ==
+<span class="exam">(b) &nbsp;<math>{\displaystyle \sum_{n=1}^{\infty}}\,(-1)^{n}\cos\frac{\pi}{n}</math>
-<span class="exam">[[009C_Sample_Midterm_3,_Problem_5|<span class="biglink">&nbsp;Problem 5.&nbsp;</span>]] &nbsp;Find the radius of convergence and the interval of convergence
+== [[009C_Sample Midterm 3,_Problem_5|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 5&nbsp;</span>]] ==
 of the series.
-::<span class="exam">(a) (6 points) &nbsp;&nbsp;&nbsp;&nbsp; <math>{\displaystyle \sum_{n=0}^{\infty}}\frac{(-1)^{n}x^{n}}{n+1}.</math>
+<span class="exam">(a) &nbsp;<math>{\displaystyle \sum_{n=0}^{\infty}}\frac{(-1)^{n}x^{n}}{n+1}</math>
-::<span class="exam">(b) (6 points) &nbsp;&nbsp;&nbsp;&nbsp; <math>{\displaystyle \sum_{n=0}^{\infty}}\frac{(x+1)^{n}}{n^{2}}.</math>
+<span class="exam">(b) &nbsp;<math>{\displaystyle \sum_{n=0}^{\infty}}\frac{(x+1)^{n}}{n^{2}}</math>
 '''Contributions to this page were made by [[Contributors|John Simanyi]]'''

← Older revision		Revision as of 18:39, 28 July 2015
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	::<span class="exam">(b) (6 points)      <math>{\displaystyle \sum_{n=0}^{\infty}}\frac{(x+1)^{n}}{n^{2}}.</math>		::<span class="exam">(b) (6 points)      <math>{\displaystyle \sum_{n=0}^{\infty}}\frac{(x+1)^{n}}{n^{2}}.</math>
		+
		+
		+	'''Contributions to this page were made by [[Contributors\|John Simanyi]]'''

← Older revision		Revision as of 21:03, 28 April 2015
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−	'''This is a department sample midterm, and is meant to represent the material usually covered in Math 9C through the midterm.'''<br>'''Click on the <span class="biglink" style="color:darkblue;"> boxed problem numbers </span> to go to a solution.'''	+	'''This is a department sample midterm, and is meant to represent the material usually covered in Math 9C through the midterm.'''<br>'''Click on the <span class="biglink" style="color:darkblue;"> boxed problem numbers </span> to go to a solution. An actual test may or may not be similar.'''

	'''In-class Instructions:''' This exam has a total of 60 points. You have 50 minutes. You must show all your work to receive full credit You may use any		'''In-class Instructions:''' This exam has a total of 60 points. You have 50 minutes. You must show all your work to receive full credit You may use any

← Older revision		Revision as of 22:25, 26 April 2015
Line 1:		Line 1:
−	'''This is a department sample midterm, and is meant to represent the material usually covered in Math 9C through the midterm. Click on the <span class="biglink" style="color:darkblue;"> boxed problem numbers </span> to go to a solution.'''	+	'''This is a department sample midterm, and is meant to represent the material usually covered in Math 9C through the midterm.'''<br>'''Click on the <span class="biglink" style="color:darkblue;"> boxed problem numbers </span> to go to a solution.'''

	'''In-class Instructions:''' This exam has a total of 60 points. You have 50 minutes. You must show all your work to receive full credit You may use any		'''In-class Instructions:''' This exam has a total of 60 points. You have 50 minutes. You must show all your work to receive full credit You may use any