009C Sample Final 3, Problem 2 - Revision history

Kayla Murray at 01:41, 3 December 2017

2017-12-03T01:41:50Z

2017-03-17T18:02:53Z

2017-03-17T17:56:48Z

2017-03-05T20:58:53Z

2017-03-05T20:56:36Z

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2017-02-18T18:42:15Z

@@ Line 7: / Line 7: @@
 <span class="exam">(b) Test if the series converges conditionally. Give reasons for your answer.
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+<hr>
-!Foundations: &nbsp;
+[[009C Sample Final 3, Problem 2 Solution|'''<u>Solution</u>''']]
-'''Solution:'''
+[[009C Sample Final 3, Problem 2 Detailed Solution|'''<u>Detailed Solution</u>''']]
 [[009C_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

@@ Line 102: / Line 102: @@
 !Final Answer: &nbsp;
 |-
-|&nbsp; &nbsp;'''(a)'''&nbsp; &nbsp; not absolutely convergent
+|&nbsp; &nbsp;'''(a)'''&nbsp; &nbsp; not absolutely convergent (by the p test)
 |-
-|&nbsp; &nbsp;'''(b)'''&nbsp; &nbsp; conditionally convergent
+|&nbsp; &nbsp;'''(b)'''&nbsp; &nbsp; conditionally convergent (by the Alternating Series Test)
 |}
 [[009C_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

@@ Line 68: / Line 68: @@
 |-
 |Let &nbsp;<math style="vertical-align: -20px"> b_n=\frac{1}{\sqrt{n}}.</math>
 |-
 |The sequence &nbsp;<math style="vertical-align: -4px">\{b_n\}</math>&nbsp; is decreasing since

@@ Line 63: / Line 63: @@
 |For
 |-
-|&nbsp; &nbsp; &nbsp; &nbsp; <math>\sum_{n=1}^\infty \frac{(-1)^n}{n},</math>
+|&nbsp; &nbsp; &nbsp; &nbsp; <math>\sum_{n=1}^\infty \frac{(-1)^n}{\sqrt{n}},</math>
 |-
 |we notice that this series is alternating.
 |-
-|Let &nbsp;<math style="vertical-align: -14px"> b_n=\frac{1}{n}.</math>
+|Let &nbsp;<math style="vertical-align: -20px"> b_n=\frac{1}{\sqrt{n}}.</math>
 |-
 |The sequence &nbsp;<math style="vertical-align: -4px">\{b_n\}</math>&nbsp; is decreasing since
 |-
-|&nbsp; &nbsp; &nbsp; &nbsp; <math>\frac{1}{n+1}<\frac{1}{n}</math>
+|&nbsp; &nbsp; &nbsp; &nbsp; <math>\frac{1}{\sqrt{n+1}}<\frac{1}{\sqrt{n}}</math>
 |-
 |for all &nbsp;<math style="vertical-align: -3px">n\ge 1.</math>
@@ Line 77: / Line 77: @@
 |Also,
 |-
-|&nbsp; &nbsp; &nbsp; &nbsp; <math>\lim_{n\rightarrow \infty}b_n=\lim_{n\rightarrow \infty}\frac{1}{n}=0.</math>
+|&nbsp; &nbsp; &nbsp; &nbsp; <math>\lim_{n\rightarrow \infty}b_n=\lim_{n\rightarrow \infty}\frac{1}{\sqrt{n}}=0.</math>
 |-
-|Therefore, the series &nbsp;<math>\sum_{n=1}^\infty \frac{(-1)^n}{n}</math> &nbsp; converges
+|Therefore, the series &nbsp;<math>\sum_{n=1}^\infty \frac{(-1)^n}{\sqrt{n}}</math> &nbsp; converges
 |-
 |by the Alternating Series Test.

@@ Line 61: / Line 61: @@
 !Step 1: &nbsp;
 |-
+|For
 |-
+|&nbsp; &nbsp; &nbsp; &nbsp; <math>\sum_{n=1}^\infty \frac{(-1)^n}{n},</math>
 |-
+|we notice that this series is alternating.
 |}
@@ Line 71: / Line 87: @@
 !Step 2: &nbsp;
 |-
+|Since the series &nbsp;<math>\sum_{n=1}^\infty \frac{(-1)^n}{\sqrt{n}}</math> &nbsp; is not absolutely convergent but convergent,
 |-
+|this series is conditionally convergent.
 |}
@@ Line 82: / Line 98: @@
 |&nbsp; &nbsp;'''(a)'''&nbsp; &nbsp; not absolutely convergent
 |-
-|&nbsp;&nbsp; '''(b)'''
+|&nbsp; &nbsp;'''(b)'''&nbsp; &nbsp; conditionally convergent
 |}
 [[009C_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

@@ Line 10: / Line 10: @@
 !Foundations: &nbsp;
 |-
+|'''1.''' A series &nbsp;<math>\sum a_n</math>&nbsp; is '''absolutely convergent''' if
 |-
+|&nbsp; &nbsp; &nbsp; &nbsp; the series &nbsp;<math>\sum |a_n|</math>&nbsp; converges.
 |-
+|'''2.''' A series &nbsp;<math>\sum a_n</math>&nbsp; is '''conditionally convergent''' if
 |-
+|&nbsp; &nbsp; &nbsp; &nbsp; the series &nbsp;<math>\sum |a_n|</math>&nbsp; diverges and the series &nbsp;<math>\sum a_n</math>&nbsp; converges.
 |}

@@ Line 1: / Line 1: @@
 <span class="exam"> Consider the series
-::::<math>\sum_{n=2}^\infty \frac{(-1)^n}{\sqrt{n}}.</math>
+::<math>\sum_{n=2}^\infty \frac{(-1)^n}{\sqrt{n}}.</math>
-::<span class="exam">a) Test if the series converges absolutely. Give reasons for your answer.
+<span class="exam">(a) Test if the series converges absolutely. Give reasons for your answer.
-::<span class="exam">b) Test if the series converges conditionally. Give reasons for your answer.
+<span class="exam">(b) Test if the series converges conditionally. Give reasons for your answer.
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
@@ Line 20: / Line 20: @@
+|
 |}
 '''Solution:'''
@@ Line 62: / Line 63: @@
+|
 |}
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"