009C Sample Midterm 2, Problem 5 - Revision history

Kayla Murray at 19:39, 12 November 2017

2017-11-12T19:39:23Z

Kayla Murray at 01:15, 5 November 2017

2017-11-05T01:15:17Z

Kayla Murray at 17:46, 18 March 2017

2017-03-18T17:46:41Z

Kayla Murray at 02:14, 27 February 2017

2017-02-27T02:14:14Z

Kayla Murray at 00:21, 19 February 2017

2017-02-19T00:21:04Z

Kayla Murray at 00:51, 16 February 2017

2017-02-16T00:51:02Z

Kayla Murray at 00:44, 16 February 2017

2017-02-16T00:44:18Z

Kayla Murray at 17:06, 13 February 2017

2017-02-13T17:06:09Z

Kayla Murray at 16:33, 5 February 2017

2017-02-05T16:33:25Z

Kayla Murray: Created page with " ::a) ::b) {| class="mw-collapsible mw-collapsed" style = "text-align:left;" !Foundations: |- | |- | :: |..."

2017-02-05T16:29:12Z

Created page with " ::a) ::b) {| class="mw-collapsible mw-collapsed" style = "text-align:left;" !Foundations: |- | |- | :: |..."

New page

::a)
::b)

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Foundations:  
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'''Solution:'''

'''(a)'''
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!Step 1:  
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'''(b)'''
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!Step 1:  
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Final Answer:  
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|'''(a)'''
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|'''(b)'''
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[[009C_Sample_Midterm_2|'''Return to Sample Exam''']]

@@ Line 7: / Line 7: @@
 <hr>
-(insert picture of handwritten solution)
+[[009C Sample Midterm 2, Problem 5 Solution|'''<u>Solution</u>''']]
 [[009C_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']]

← Older revision		Revision as of 01:15, 5 November 2017
Line 6:		Line 6:


		+	<hr>
		+	(insert picture of handwritten solution)

−	{\| ~~class="mw-collapsible mw-collapsed" style = "text-align:left;"~~	+	[[009C Sample Midterm 2, Problem 5 Detailed Solution\|'''<u>Detailed Solution for this Problem</u>''']]
−	~~!Foundations:  ~~
−	\|-
−	~~\|A geometric series  ~~<~~math~~>~~\sum_{n=0}^{\infty} ar^n~~</~~math>  converges if  <math style="vertical-align: -6px"~~>~~\|r\|<1.</math>~~
−	\|}

−
−	~~'''Solution:'''~~
−
−	~~'''(a)'''~~
−	~~{\| class="mw-collapsible mw-collapsed" style = "text-align:left;"~~
−	~~!Step 1:  ~~
−	\|-
−	~~\|First, we notice that  <math>\sum_{n=0}^\infty c_nx^n</math>  is a geometric series.~~
−	\|-
−	~~\|We have  <math style="vertical-align: -1px">r=x.</math>~~
−	\|-
−	~~\|Since this series converges,~~
−	\|-
−	~~\|        <math>\|r\|=\|x\|<1.</math>~~
−	\|}
−
−	~~{\| class="mw-collapsible mw-collapsed" style = "text-align:left;"~~
−	~~!Step 2:  ~~
−	\|-
−	~~\|The series  <math>\sum_{n=0} c_n\bigg(\frac{x}{2}\bigg)^n</math>  is also a geometric series.~~
−	\|-
−	~~\|For this series,  <math style="vertical-align: -13px">r=\frac{x}{2}.</math>~~
−	\|-
−	~~\|Now, we notice~~
−	\|-
−	\|
−	~~        <math>\begin{array}{rcl}~~
−	~~\displaystyle{\|r\|} & = & \displaystyle{\bigg\|\frac{x}{2}\bigg\|}\\~~
−	~~&&\\~~
−	~~& = & \displaystyle{\frac{\|x\|}{2}}\\~~
−	~~&&\\~~
−	~~& < & \displaystyle{\frac{1}{2}}~~
−	~~\end{array}</math>~~
−	\|-
−	~~\|since  <math style="vertical-align: -5px">\|x\|<1.</math>~~
−	\|-
−	~~\| Since  <math style="vertical-align: -5px">\|r\|<1,</math>  this series converges.~~
−	\|}
−
−	~~'''(b)'''~~
−	~~{\| class="mw-collapsible mw-collapsed" style = "text-align:left;"~~
−	~~!Step 1:  ~~
−	\|-
−	~~\|First, we notice that  <math>\sum_{n=0}^\infty c_nx^n</math>  is a geometric series.~~
−	\|-
−	~~\|We have  <math style="vertical-align: -1px">r=x.</math>~~
−	\|-
−	~~\|Since this series converges,~~
−	\|-
−	~~\|        <math>\|r\|=\|x\|<1.</math>~~
−	\|}
−
−	~~{\| class="mw-collapsible mw-collapsed" style = "text-align:left;"~~
−	~~!Step 2:  ~~
−	\|-
−	~~\|The series  <math>\sum_{n=0}^\infty c_n(-x)^n</math>  is also a geometric series.~~
−	\|-
−	~~\|For this series,  <math style="vertical-align: -1px">r=-x.</math>~~
−	\|-
−	~~\|Now, we notice~~
−	\|-
−	\|
−	~~        <math>\begin{array}{rcl}~~
−	~~\displaystyle{\|r\|} & = & \displaystyle{\|-x\|}\\~~
−	~~&&\\~~
−	~~& = & \displaystyle{\|x\|}\\~~
−	~~&&\\~~
−	~~& < & \displaystyle{1}~~
−	~~\end{array}</math>~~
−	\|-
−	~~\|since  <math style="vertical-align: -5px">\|x\|<1.</math>~~
−	\|-
−	~~\|Since  <math style="vertical-align: -5px">\|r\|<1,</math>  this series converges.~~
−	\|}
−
−
−	~~{\| class="mw-collapsible mw-collapsed" style = "text-align:left;"~~
−	~~!Final Answer:  ~~
−	\|-
−	~~\|    '''(a)'''     converges (by the geometric series test)~~
−	\|-
−	~~\|    '''(b)'''     converges (by the geometric series test)~~
−	\|}
	[[009C_Sample_Midterm_2\|'''<u>Return to Sample Exam</u>''']]		[[009C_Sample_Midterm_2\|'''<u>Return to Sample Exam</u>''']]

@@ Line 92: / Line 92: @@
 !Final Answer: &nbsp;
 |-
-|&nbsp; &nbsp; '''(a)''' &nbsp; &nbsp; The series converges.
+|&nbsp; &nbsp; '''(a)''' &nbsp; &nbsp; converges (by the geometric series test)
 |-
-|&nbsp; &nbsp; '''(b)''' &nbsp; &nbsp; The series converges.
+|&nbsp; &nbsp; '''(b)''' &nbsp; &nbsp; converges (by the geometric series test)
 |}
 [[009C_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']]

@@ Line 1: / Line 1: @@
-<span class="exam">If <math>\sum_{n=0}^\infty c_nx^n</math> converges, does it follow that the following series converges?
+<span class="exam">If &nbsp;<math>\sum_{n=0}^\infty c_nx^n</math>&nbsp; converges, does it follow that the following series converges?
-<span class="exam">(a) <math>\sum_{n=0}^\infty c_n\bigg(\frac{x}{2}\bigg)^n</math>
+<span class="exam">(a) &nbsp;<math>\sum_{n=0}^\infty c_n\bigg(\frac{x}{2}\bigg)^n</math>
-<span class="exam">(b) <math>\sum_{n=0}^\infty c_n(-x)^n </math>
+<span class="exam">(b) &nbsp;<math>\sum_{n=0}^\infty c_n(-x)^n </math>
@@ Line 10: / Line 10: @@
 !Foundations: &nbsp;
 |-
-|A geometric series <math>\sum_{n=0}^{\infty} ar^n</math> converges if <math style="vertical-align: -6px">|r|<1.</math>
+|A geometric series &nbsp;<math>\sum_{n=0}^{\infty} ar^n</math>&nbsp; converges if &nbsp;<math style="vertical-align: -6px">|r|<1.</math>
 |}
@@ Line 20: / Line 20: @@
 !Step 1: &nbsp;
 |-
-|First, we notice that <math>\sum_{n=0}^\infty c_nx^n</math> is a geometric series.
+|First, we notice that &nbsp;<math>\sum_{n=0}^\infty c_nx^n</math>&nbsp; is a geometric series.
 |-
-|We have <math style="vertical-align: -1px">r=x.</math>
+|We have &nbsp;<math style="vertical-align: -1px">r=x.</math>
 |-
 |Since this series converges,
@@ Line 32: / Line 32: @@
 !Step 2: &nbsp;
 |-
-|The series <math>\sum_{n=0} c_n\bigg(\frac{x}{2}\bigg)^n</math> is also a geometric series.
+|The series &nbsp;<math>\sum_{n=0} c_n\bigg(\frac{x}{2}\bigg)^n</math>&nbsp; is also a geometric series.
 |-
-|For this series, <math style="vertical-align: -13px">r=\frac{x}{2}.</math>
+|For this series, &nbsp;<math style="vertical-align: -13px">r=\frac{x}{2}.</math>
 |-
 |Now, we notice
@@ Line 47: / Line 47: @@
 \end{array}</math>
 |-
-|since <math style="vertical-align: -5px">|x|<1.</math>
+|since &nbsp;<math style="vertical-align: -5px">|x|<1.</math>
 |-
-| Since <math style="vertical-align: -5px">|r|<1,</math> this series converges.
+| Since &nbsp;<math style="vertical-align: -5px">|r|<1,</math>&nbsp; this series converges.
 |}
@@ Line 56: / Line 56: @@
 !Step 1: &nbsp;
 |-
-|First, we notice that <math>\sum_{n=0}^\infty c_nx^n</math> is a geometric series.
+|First, we notice that &nbsp;<math>\sum_{n=0}^\infty c_nx^n</math>&nbsp; is a geometric series.
 |-
-|We have <math style="vertical-align: -1px">r=x.</math>
+|We have &nbsp;<math style="vertical-align: -1px">r=x.</math>
 |-
 |Since this series converges,
@@ Line 68: / Line 68: @@
 !Step 2: &nbsp;
 |-
-|The series <math>\sum_{n=0}^\infty c_n(-x)^n</math> is also a geometric series.
+|The series &nbsp;<math>\sum_{n=0}^\infty c_n(-x)^n</math>&nbsp; is also a geometric series.
 |-
-|For this series, <math style="vertical-align: -1px">r=-x.</math>
+|For this series, &nbsp;<math style="vertical-align: -1px">r=-x.</math>
 |-
 |Now, we notice
@@ Line 83: / Line 83: @@
 \end{array}</math>
 |-
-|since <math style="vertical-align: -5px">|x|<1.</math>
+|since &nbsp;<math style="vertical-align: -5px">|x|<1.</math>
 |-
-|Since <math style="vertical-align: -5px">|r|<1,</math> this series converges.
+|Since &nbsp;<math style="vertical-align: -5px">|r|<1,</math>&nbsp; this series converges.
 |}

@@ Line 1: / Line 1: @@
 <span class="exam">If <math>\sum_{n=0}^\infty c_nx^n</math> converges, does it follow that the following series converges?
-::<span class="exam">a) <math>\sum_{n=0}^\infty c_n\bigg(\frac{x}{2}\bigg)^n</math>
+<span class="exam">(a) <math>\sum_{n=0}^\infty c_n\bigg(\frac{x}{2}\bigg)^n</math>
-::<span class="exam">b) <math>\sum_{n=0}^\infty c_n(-x)^n </math>

@@ Line 9: / Line 9: @@
 !Foundations: &nbsp;
 |-
-| Geometric Series
+|A geometric series <math>\sum_{n=0}^{\infty} ar^n</math> converges if <math>|r|<1.</math>
 |}

← Older revision		Revision as of 16:33, 5 February 2017
Line 1:		Line 1:
−	<span class="exam">	+	<span class="exam">If <math>\sum_{n=0}^\infty c_nx^n</math> converges, does it follow that the following series converges?
		+
		+	::<span class="exam">a) <math>\sum_{n=0}^\infty c_n\bigg(\frac{x}{2}\bigg)^n</math>
		+	::<span class="exam">b) <math>\sum_{n=0}^\infty c_n(-x)^n </math>

−	~~::<span class="exam">a)~~
−	~~::<span class="exam">b)~~