009C Sample Midterm 2, Problem 2 - Revision history

Kayla Murray at 19:34, 12 November 2017

2017-11-12T19:34:11Z

Kayla Murray at 01:11, 5 November 2017

2017-11-05T01:11:58Z

Kayla Murray at 17:34, 18 March 2017

2017-03-18T17:34:06Z

Kayla Murray at 02:05, 27 February 2017

2017-02-27T02:05:26Z

Kayla Murray at 19:07, 26 February 2017

2017-02-26T19:07:17Z

Kayla Murray at 00:19, 19 February 2017

2017-02-19T00:19:10Z

Kayla Murray at 00:02, 16 February 2017

2017-02-16T00:02:19Z

Kayla Murray at 23:57, 15 February 2017

2017-02-15T23:57:59Z

Kayla Murray at 18:06, 13 February 2017

2017-02-13T18:06:57Z

Kayla Murray at 16:31, 5 February 2017

2017-02-05T16:31:47Z

@@ Line 4: / Line 4: @@
 <hr>
-(insert picture of handwritten solution)
+[[009C Sample Midterm 2, Problem 2 Solution|'''<u>Solution</u>''']]
 [[009C_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']]

← Older revision		Revision as of 01:11, 5 November 2017
Line 3:		Line 3:
	::<math>\sum_{n=1}^\infty \frac{3^n}{n}</math>		::<math>\sum_{n=1}^\infty \frac{3^n}{n}</math>

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

		+	[[009C Sample Midterm 2, Problem 2 Detailed Solution\|'''<u>Detailed Solution for this Problem</u>''']]

−	~~{\| class="mw-collapsible mw-collapsed" style = "text-align:left;"~~
−	~~!Foundations:  ~~
−	\|-
−	~~\|'''Direct Comparison Test'''~~
−	\|-
−	~~\|        Let  <math>\{a_n\}</math>  and  <math>\{b_n\}</math>  be positive sequences where  <math style="vertical-align: -3px">a_n\le b_n</math>~~
−	\|-
−	~~\|        for all  <math style="vertical-align: -3px">n\ge N</math>  for some  <math style="vertical-align: -3px">N\ge 1.</math>~~
−	\|-
−	~~\|        '''1.''' If  <math>\sum_{n=1}^\infty b_n</math>  converges, then  <math>\sum_{n=1}^\infty a_n</math>  converges.~~
−	\|-
−	~~\|        '''2.''' If  <math>\sum_{n=1}^\infty a_n</math>  diverges, then  <math>\sum_{n=1}^\infty b_n</math>  diverges.~~
−	\|}
−
−
−	~~'''Solution:'''~~
−
−	~~{\| class="mw-collapsible mw-collapsed" style = "text-align:left;"~~
−	~~!Step 1:  ~~
−	\|-
−	~~\|First, we note that~~
−	\|-
−	~~\|        <math>\frac{3^n}{n}>0</math>~~
−	\|-
−	~~\|for all  <math style="vertical-align: -3px">n\ge 1.</math>~~
−	\|-
−	~~\|This means that we can use a comparison test on this series.~~
−	\|-
−	~~\|Let  <math style="vertical-align: -13px">a_n=\frac{3^n}{n}.</math>~~
−	\|}
−
−	~~{\| class="mw-collapsible mw-collapsed" style = "text-align:left;"~~
−	~~!Step 2:  ~~
−	\|-
−	~~\|Let  <math style="vertical-align: -14px">b_n=\frac{1}{n}.</math>~~
−	\|-
−	~~\|We want to compare the series in this problem with~~
−	\|-
−	~~\|        <math>\sum_{n=1}^\infty \frac{1}{n}.</math>~~
−	\|-
−	~~\|This is the harmonic series (or  <math style="vertical-align: -4px">p</math>-series with  <math style="vertical-align: -4px">p=1.</math> )~~
−	\|-
−	~~\|Hence,  <math>\sum_{n=1}^\infty b_n</math>  diverges.~~
−	\|}
−
−	~~{\| class="mw-collapsible mw-collapsed" style = "text-align:left;"~~
−	~~!Step 3:  ~~
−	\|-
−	~~\|Also, we have  <math style="vertical-align: -4px">b_n<a_n</math>  since~~
−	\|-
−	~~\|        <math>\frac{1}{n}<\frac{3^n}{n}</math>~~
−	\|-
−	~~\| for all  <math style="vertical-align: -3px">n\ge 1.</math>~~
−	\|-
−	~~\|Therefore, the series  <math>\sum_{n=1}^\infty a_n</math>  diverges~~
−	\|-
−	~~\|by the Direct Comparison Test.~~
−	\|}
−
−
−	~~{\| class="mw-collapsible mw-collapsed" style = "text-align:left;"~~
−	~~!Final Answer:  ~~
−	\|-
−	~~\|        diverges (by the Direct Comparison Test)~~
−	\|}
	[[009C_Sample_Midterm_2\|'''<u>Return to Sample Exam</u>''']]		[[009C_Sample_Midterm_2\|'''<u>Return to Sample Exam</u>''']]

@@ Line 8: / Line 8: @@
 !Foundations: &nbsp;
 |-
-|'''1.''' '''Direct Comparison Test'''
+|'''Direct Comparison Test'''
 |-
 |&nbsp; &nbsp; &nbsp; &nbsp; Let &nbsp;<math>\{a_n\}</math>&nbsp; and &nbsp;<math>\{b_n\}</math>&nbsp; be positive sequences where &nbsp;<math style="vertical-align: -3px">a_n\le b_n</math>
@@ Line 14: / Line 14: @@
 |&nbsp; &nbsp; &nbsp; &nbsp; for all &nbsp;<math style="vertical-align: -3px">n\ge N</math>&nbsp; for some &nbsp;<math style="vertical-align: -3px">N\ge 1.</math>
 |-
-|'''2.''' If &nbsp;<math>\sum_{n=1}^\infty b_n</math>&nbsp; converges, then &nbsp;<math>\sum_{n=1}^\infty a_n</math>&nbsp; converges.
+|&nbsp; &nbsp; &nbsp; &nbsp; '''1.''' If &nbsp;<math>\sum_{n=1}^\infty b_n</math>&nbsp; converges, then &nbsp;<math>\sum_{n=1}^\infty a_n</math>&nbsp; converges.
 |-
-|'''3.''' If &nbsp;<math>\sum_{n=1}^\infty a_n</math>&nbsp; diverges, then &nbsp;<math>\sum_{n=1}^\infty b_n</math>&nbsp; diverges.
+|&nbsp; &nbsp; &nbsp; &nbsp; '''2.''' If &nbsp;<math>\sum_{n=1}^\infty a_n</math>&nbsp; diverges, then &nbsp;<math>\sum_{n=1}^\infty b_n</math>&nbsp; diverges.
 |}
@@ Line 68: / Line 68: @@
 !Final Answer: &nbsp;
 |-
-|&nbsp; &nbsp; &nbsp; &nbsp; diverges
+|&nbsp; &nbsp; &nbsp; &nbsp; diverges (by the Direct Comparison Test)
 |}
 [[009C_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']]

@@ Line 10: / Line 10: @@
 |'''1.''' '''Direct Comparison Test'''
 |-
-|&nbsp; &nbsp; &nbsp; &nbsp; Let <math>\{a_n\}</math> and <math>\{b_n\}</math> be positive sequences where <math style="vertical-align: -3px">a_n\le b_n</math>
+|&nbsp; &nbsp; &nbsp; &nbsp; Let &nbsp;<math>\{a_n\}</math>&nbsp; and &nbsp;<math>\{b_n\}</math>&nbsp; be positive sequences where &nbsp;<math style="vertical-align: -3px">a_n\le b_n</math>
 |-
-|&nbsp; &nbsp; &nbsp; &nbsp; for all <math style="vertical-align: -3px">n\ge N</math> for some <math style="vertical-align: -3px">N\ge 1.</math>
+|&nbsp; &nbsp; &nbsp; &nbsp; for all &nbsp;<math style="vertical-align: -3px">n\ge N</math>&nbsp; for some &nbsp;<math style="vertical-align: -3px">N\ge 1.</math>
 |-
-|'''2.''' If <math>\sum_{n=1}^\infty b_n</math> converges, then <math>\sum_{n=1}^\infty a_n</math> converges.
+|'''2.''' If &nbsp;<math>\sum_{n=1}^\infty b_n</math>&nbsp; converges, then &nbsp;<math>\sum_{n=1}^\infty a_n</math>&nbsp; converges.
 |-
-|'''3.''' If <math>\sum_{n=1}^\infty a_n</math> diverges, then <math>\sum_{n=1}^\infty b_n</math> diverges.
+|'''3.''' If &nbsp;<math>\sum_{n=1}^\infty a_n</math>&nbsp; diverges, then &nbsp;<math>\sum_{n=1}^\infty b_n</math>&nbsp; diverges.
 |}
@@ Line 29: / Line 29: @@
 |&nbsp; &nbsp; &nbsp; &nbsp; <math>\frac{3^n}{n}>0</math>
 |-
-|for all <math style="vertical-align: -3px">n\ge 1.</math>
+|for all &nbsp;<math style="vertical-align: -3px">n\ge 1.</math>
 |-
 |This means that we can use a comparison test on this series.
 |-
-|Let <math style="vertical-align: -13px">a_n=\frac{3^n}{n}.</math>
+|Let &nbsp;<math style="vertical-align: -13px">a_n=\frac{3^n}{n}.</math>
 |}
@@ Line 39: / Line 39: @@
 !Step 2: &nbsp;
 |-
-|Let <math style="vertical-align: -14px">b_n=\frac{1}{n}.</math>
+|Let &nbsp;<math style="vertical-align: -14px">b_n=\frac{1}{n}.</math>
 |-
 |We want to compare the series in this problem with
@@ Line 45: / Line 45: @@
 |&nbsp; &nbsp; &nbsp; &nbsp; <math>\sum_{n=1}^\infty \frac{1}{n}.</math>
 |-
-|This is the harmonic series (or <math style="vertical-align: -4px">p</math>-series with <math style="vertical-align: -4px">p=1.</math>)
+|This is the harmonic series (or &nbsp;<math style="vertical-align: -4px">p</math>-series with &nbsp;<math style="vertical-align: -4px">p=1.</math>&nbsp;)
 |-
-|Hence, <math>\sum_{n=1}^\infty b_n</math> diverges.
+|Hence, &nbsp;<math>\sum_{n=1}^\infty b_n</math>&nbsp; diverges.
 |}
@@ Line 53: / Line 53: @@
 !Step 3: &nbsp;
 |-
-|Also, we have <math style="vertical-align: -4px">b_n<a_n</math> since
+|Also, we have &nbsp;<math style="vertical-align: -4px">b_n<a_n</math>&nbsp; since
 |-
 |&nbsp; &nbsp; &nbsp; &nbsp; <math>\frac{1}{n}<\frac{3^n}{n}</math>
 |-
-| for all <math style="vertical-align: -3px">n\ge 1.</math>
+| for all &nbsp;<math style="vertical-align: -3px">n\ge 1.</math>
 |-
-|Therefore, the series <math>\sum_{n=1}^\infty a_n</math> diverges
+|Therefore, the series &nbsp;<math>\sum_{n=1}^\infty a_n</math>&nbsp; diverges
 |-
 |by the Direct Comparison Test.

@@ Line 8: / Line 8: @@
 !Foundations: &nbsp;
 |-
-|'''Direct Comparison Test'''
+|'''1.''' '''Direct Comparison Test'''
 |-
 |&nbsp; &nbsp; &nbsp; &nbsp; Let <math>\{a_n\}</math> and <math>\{b_n\}</math> be positive sequences where <math style="vertical-align: -3px">a_n\le b_n</math>
@@ Line 14: / Line 14: @@
 |&nbsp; &nbsp; &nbsp; &nbsp; for all <math style="vertical-align: -3px">n\ge N</math> for some <math style="vertical-align: -3px">N\ge 1.</math>
 |-
-|&nbsp; &nbsp; &nbsp; &nbsp; '''1.''' If <math>\sum_{n=1}^\infty b_n</math> converges, then <math>\sum_{n=1}^\infty a_n</math> converges.
+|'''2.''' If <math>\sum_{n=1}^\infty b_n</math> converges, then <math>\sum_{n=1}^\infty a_n</math> converges.
 |-
-|&nbsp; &nbsp; &nbsp; &nbsp; '''2.''' If <math>\sum_{n=1}^\infty a_n</math> diverges, then <math>\sum_{n=1}^\infty b_n</math> diverges.
+|'''3.''' If <math>\sum_{n=1}^\infty a_n</math> diverges, then <math>\sum_{n=1}^\infty b_n</math> diverges.
 |}
 '''Solution:'''

← Older revision		Revision as of 00:19, 19 February 2017
Line 1:		Line 1:
	<span class="exam">Determine convergence or divergence:		<span class="exam">Determine convergence or divergence:

−	~~::::~~::<math>\sum_{n=1}^\infty \frac{3^n}{n}</math>	+	::<math>\sum_{n=1}^\infty \frac{3^n}{n}</math>

@@ Line 8: / Line 8: @@
 !Foundations: &nbsp;
 |-
+| Direct Comparison Test
 |-
+|
 |-
+|
 |}
@@ Line 22: / Line 20: @@
 !Step 1: &nbsp;
 |-
+|First, we note that
 |-
+|Let <math>a_n=\frac{3^n}{n}.</math>
 |}
@@ Line 30: / Line 34: @@
 !Step 2: &nbsp;
 |-
+|Let <math>b_n=\frac{1}{n}.</math>
 |-
+|Hence, <math>\sum_{n=1}^\infty b_n</math> diverges.
 |}
@@ Line 40: / Line 63: @@
 !Final Answer: &nbsp;
 |-
+|&nbsp; &nbsp; &nbsp; &nbsp; diverges
 |}
 [[009C_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']]

@@ Line 1: / Line 1: @@
-<span class="exam">
+<span class="exam">Determine convergence or divergence:
@@ Line 19: / Line 19: @@
 '''Solution:'''
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Step 1: &nbsp;
@@ Line 36: / Line 35: @@
 |}
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Final Answer: &nbsp;
 |-
-|'''(a)'''
 |-
-|'''(b)'''
 |}
 [[009C_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']]