009C Sample Final 2, Problem 3 - Revision history

Kayla Murray at 01:24, 3 December 2017

2017-12-03T01:24:49Z

Kayla Murray at 18:16, 17 March 2017

2017-03-17T18:16:34Z

Kayla Murray at 18:15, 17 March 2017

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Grad at 16:55, 12 March 2017

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Grad at 00:19, 11 March 2017

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Kayla Murray at 03:50, 5 March 2017

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Kayla Murray at 03:42, 5 March 2017

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Kayla Murray at 03:40, 5 March 2017

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Kayla Murray at 00:47, 5 March 2017

2017-03-05T00:47:28Z

Kayla Murray at 00:40, 5 March 2017

2017-03-05T00:40:16Z

@@ Line 5: / Line 5: @@
 <span class="exam">(b) &nbsp;<math>\sum_{n=1}^{\infty} (-1)^n\frac{1}{n+1}</math>
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+<hr>
-!Foundations: &nbsp;
+[[009C Sample Final 2, Problem 3 Solution|'''<u>Solution</u>''']]
-'''Solution:'''
+[[009C Sample Final 2, Problem 3 Detailed Solution|'''<u>Detailed Solution</u>''']]
 [[009C_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]

@@ Line 1: / Line 1: @@
-<span class="exam">Determine if the following series converges or diverges. Please give your reason(s).
+<span class="exam"> Determine if the following series converges or diverges. Please give your reason(s).
 <span class="exam">(a) &nbsp;<math>\sum_{n=1}^{\infty} \frac{n!}{(2n)!}</math>
-<span class="exam">(b) &nbsp;<math>\sum_{n=1}^{\infty} (-1)^n \frac{1}{n+1}</math>
+<span class="exam">(b) &nbsp;<math>\sum_{n=1}^{\infty} (-1)^n\frac{1}{n+1}</math>
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
@@ Line 48: / Line 48: @@
+|
 &nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
-\displaystyle{\lim_{n\rightarrow \infty}\bigg|\frac{a_{n+1}}{a_n}\bigg|} & = & \displaystyle{\lim_{n\rightarrow \infty} \bigg| \frac{(n+1)!}{(2(n+1))!} \cdot\frac{(2n)!}{n!}\bigg|}\\
+\displaystyle{\lim_{n\rightarrow \infty}\bigg|\frac{a_{n+1}}{a_n}\bigg|} & = & \displaystyle{\lim_{n\rightarrow \infty} \bigg| \frac{(n+1)!}{(2(n+1))!} \frac{(2n)!}{n!}\bigg|}\\
 &&\\
-& = & \displaystyle{\lim_{n\rightarrow \infty} \bigg| \frac{(n+1)n!}{(2n+2)(2n+1)(2n)!} \cdot \frac{(2n)!}{n!}\bigg|}\\
+& = & \displaystyle{\lim_{n\rightarrow \infty} \bigg| \frac{(n+1)n!}{(2n+2)(2n+1)(2n)!} \frac{(2n)!}{n!}\bigg|}\\
 &&\\
 & = & \displaystyle{\lim_{n\rightarrow \infty} \frac{n+1}{(2n+2)(2n+1)}}\\
@@ Line 82: / Line 82: @@
 |-
 |Let &nbsp;<math style="vertical-align: -16px"> b_n=\frac{1}{n+1}.</math>
 |-
 |The sequence &nbsp;<math style="vertical-align: -5px">\{b_n\}</math>&nbsp; is decreasing since
@@ Line 87: / Line 93: @@
 |&nbsp; &nbsp; &nbsp; &nbsp; <math>\frac{1}{n+2}<\frac{1}{n+1}</math>
 |-
-|for all &nbsp;<math style="vertical-align: -3px">n\ge 0.</math>
+|for all &nbsp;<math style="vertical-align: -3px">n\ge 1.</math>
 |}
@@ Line 106: / Line 112: @@
 !Final Answer: &nbsp;
 |-
-|&nbsp;&nbsp; '''(a)''' &nbsp;&nbsp; converges
+|&nbsp;&nbsp; '''(a)''' &nbsp;&nbsp; converges (by the Ratio Test)
 |-
-|&nbsp;&nbsp; '''(b)''' &nbsp;&nbsp; converges
+|&nbsp;&nbsp; '''(b)''' &nbsp;&nbsp; converges (by the Alternating Series Test)
 |}
 [[009C_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]

@@ Line 1: / Line 1: @@
 <span class="exam">Determine if the following series converges or diverges. Please give your reason(s).
-<span class="exam">(a) &nbsp;<math>\sum_{n=0}^{\infty} \frac{n!}{(2n)!}</math>
+<span class="exam">(a) &nbsp;<math>\sum_{n=1}^{\infty} \frac{n!}{(2n)!}</math>
-<span class="exam">(b) &nbsp;<math>\sum_{n=0}^{\infty} (-1)^n \frac{1}{n+1}</math>
+<span class="exam">(b) &nbsp;<math>\sum_{n=1}^{\infty} (-1)^n \frac{1}{n+1}</math>
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"

@@ Line 1: / Line 1: @@
 <span class="exam">Determine if the following series converges or diverges. Please give your reason(s).
-<span class="exam">(a) &nbsp;<math>\sum_{n=0}^{+\infty} \frac{n!}{(2n)!}</math>
+<span class="exam">(a) &nbsp;<math>\sum_{n=0}^{\infty} \frac{n!}{(2n)!}</math>
-<span class="exam">(b) &nbsp;<math>\sum_{n=0}^{+\infty} (-1)^n \frac{1}{n+1}</math>
+<span class="exam">(b) &nbsp;<math>\sum_{n=0}^{\infty} (-1)^n \frac{1}{n+1}</math>
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"

@@ Line 48: / Line 48: @@
+|
 &nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
-\displaystyle{\lim_{n\rightarrow \infty}\bigg|\frac{a_{n+1}}{a_n}\bigg|} & = & \displaystyle{\lim_{n\rightarrow \infty} \bigg| \frac{(n+1)!}{(2(n+1))!} \frac{(2n)!}{n!}\bigg|}\\
+\displaystyle{\lim_{n\rightarrow \infty}\bigg|\frac{a_{n+1}}{a_n}\bigg|} & = & \displaystyle{\lim_{n\rightarrow \infty} \bigg| \frac{(n+1)!}{(2(n+1))!} \cdot\frac{(2n)!}{n!}\bigg|}\\
 &&\\
-& = & \displaystyle{\lim_{n\rightarrow \infty} \bigg| \frac{(n+1)n!}{(2n+2)(2n+1)(2n)!} \frac{(2n)!}{n!}\bigg|}\\
+& = & \displaystyle{\lim_{n\rightarrow \infty} \bigg| \frac{(n+1)n!}{(2n+2)(2n+1)(2n)!} \cdot \frac{(2n)!}{n!}\bigg|}\\
 &&\\
 & = & \displaystyle{\lim_{n\rightarrow \infty} \frac{n+1}{(2n+2)(2n+1)}}\\

@@ Line 42: / Line 42: @@
 !Step 1: &nbsp;
 |-
+|We begin by using the Ratio Test.
 |-
+|We have
 |-
+|
 |}
@@ Line 52: / Line 61: @@
 !Step 2: &nbsp;
 |-
+|Since
 |-
+|Therefore, the series converges.
 |}
@@ Line 93: / Line 106: @@
 !Final Answer: &nbsp;
 |-
-|&nbsp;&nbsp; '''(a)'''
+|&nbsp;&nbsp; '''(a)''' &nbsp;&nbsp; converges
 |-
 |&nbsp;&nbsp; '''(b)''' &nbsp;&nbsp; converges
 |}
 [[009C_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]

@@ Line 8: / Line 8: @@
 !Foundations: &nbsp;
 |-
+|'''1.''' '''Ratio Test'''
 |-
+|&nbsp; &nbsp; &nbsp; &nbsp; Then,
 |-
+|
 |-
+|
 |-
+|
 |}