Difference between revisions of "009C Sample Midterm 3"

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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;">&nbsp;boxed problem numbers&nbsp;</span> to go to a solution.'''  
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'''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
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'''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.
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<div class="noautonum">__TOC__</div>
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>
 
  
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== [[009C_Sample Midterm 3,_Problem_1|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 1&nbsp;</span></span>]] ==
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<span class="exam">Test if the following sequence <math  style="vertical-align: -10%">{a_n}</math> converges or diverges.
  
== Convergence and Limits of a Sequence ==
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<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.
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::<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>
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== [[009C_Sample Midterm 3,_Problem_2|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 2&nbsp;</span>]] ==
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<span class="exam">For each the following series find the sum, if it converges.
  
== Sum of a Series ==
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<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
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<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>  
you think it diverges, explain why.
 
 
 
::<span class="exam">(a) (6 points) &nbsp;&nbsp;&nbsp;&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>
 
<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>
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<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 ==
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== [[009C_Sample Midterm 3,_Problem_3|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 3&nbsp;</span>]] ==
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<span class="exam">Test if each the following series converges or diverges.
  
<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
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<span class="exam">Give reasons and clearly state if you are using any standard test.
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>
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<span class="exam">(a) &nbsp;<math>{\displaystyle \sum_{n=1}^{\infty}}\,\frac{n!}{(3n+1)!}</math>
 
<br>
 
<br>
  
::<span class="exam">(b) (6 points) &nbsp;&nbsp;&nbsp;&nbsp; <math>{\displaystyle \sum_{n=2}^{\infty}}\,\frac{\sqrt{n}}{n^{2}-3}.</math>
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<span class="exam">(b) &nbsp;<math>{\displaystyle \sum_{n=2}^{\infty}}\,\frac{\sqrt{n}}{n^{2}-3}</math>
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== [[009C_Sample Midterm 3,_Problem_4|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 4&nbsp;</span>]] ==
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<span class="exam">Test the series for convergence or divergence.
  
== Convergence Tests for Series II ==
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<span class="exam">(a) &nbsp;<math>{\displaystyle \sum_{n=1}^{\infty}}\,(-1)^{n}\sin\frac{\pi}{n}</math>
<span class="exam">[[009C_Sample_Midterm_3,_Problem_4|<span class="biglink">&nbsp;Problem 4.&nbsp;</span>]] &nbsp;Test the series for convergence or divergence.
 
  
::<span class="exam">(a) (6 points) &nbsp;&nbsp;&nbsp;&nbsp; <math>{\displaystyle \sum_{n=1}^{\infty}}\,(-1)^{n}\sin\frac{\pi}{n}.</math>
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<span class="exam">(b) &nbsp;<math>{\displaystyle \sum_{n=1}^{\infty}}\,(-1)^{n}\cos\frac{\pi}{n}</math>
  
::<span class="exam">(b) (6 points) &nbsp;&nbsp;&nbsp;&nbsp; <math>{\displaystyle \sum_{n=1}^{\infty}}\,(-1)^{n}\cos\frac{\pi}{n}.</math>
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== [[009C_Sample Midterm 3,_Problem_5|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 5&nbsp;</span>]] ==
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<span class="exam"> Find the radius of convergence and the interval of convergence
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of the series.
  
== Radius and Interval of Convergence ==
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<span class="exam">(a) &nbsp;<math>{\displaystyle \sum_{n=0}^{\infty}}\frac{(-1)^{n}x^{n}}{n+1}</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
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<span class="exam">(b) &nbsp;<math>{\displaystyle \sum_{n=0}^{\infty}}\frac{(x+1)^{n}}{n^{2}}</math>
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">(b) (6 points) &nbsp;&nbsp;&nbsp;&nbsp; <math>{\displaystyle \sum_{n=0}^{\infty}}\frac{(x+1)^{n}}{n^{2}}.</math>
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'''Contributions to this page were made by [[Contributors|John Simanyi]]'''

Latest revision as of 09:40, 28 November 2017

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.

Click on the  boxed problem numbers  to go to a solution.

 Problem 1 

Test if the following sequence converges or diverges.

If it converges, also find the limit of the sequence.

 Problem 2 

For each the following series find the sum, if it converges.

If you think it diverges, explain why.

(a)  

(b)  

 Problem 3 

Test if each the following series converges or diverges.

Give reasons and clearly state if you are using any standard test.

(a)  

(b)  

 Problem 4 

Test the series for convergence or divergence.

(a)  

(b)  

 Problem 5 

Find the radius of convergence and the interval of convergence of the series.

(a)  

(b)  


Contributions to this page were made by John Simanyi