009C Sample Final 3, Problem 4 - Revision history

Kayla Murray at 01:45, 3 December 2017

2017-12-03T01:45:34Z

2017-03-18T19:29:04Z

2017-03-17T18:04:17Z

2017-03-17T17:58:24Z

2017-03-12T20:56:48Z

2017-03-05T20:28:29Z

2017-03-05T00:54:37Z

2017-02-18T18:48:48Z

@@ 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 3, Problem 4 Solution|'''<u>Solution</u>''']]
-'''Solution:'''
+[[009C Sample Final 3, Problem 4 Detailed Solution|'''<u>Detailed Solution</u>''']]
 [[009C_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

@@ Line 103: / Line 103: @@
 |&nbsp; &nbsp; &nbsp; &nbsp; <math>\lim_{n\rightarrow \infty}b_n=\lim_{n\rightarrow \infty}\frac{1}{n+1}=0.</math>
 |-
-|Therefore, the series &nbsp;<math>\sum_{n=1}^\infty (-1)^n\frac{1}{n+1}</math> &nbsp; converges
+|Therefore,
 |-
-|by the Alternating Series Test.
+|&nbsp; &nbsp; &nbsp; &nbsp;<math>\sum_{n=1}^\infty (-1)^n\frac{1}{n+1}</math> &nbsp;
 |}

@@ Line 112: / 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_3|'''<u>Return to Sample Exam</u>''']]

@@ 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 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=1}^{+\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=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 8: / Line 8: @@
 !Foundations: &nbsp;
 |-
+|'''1.''' '''Ratio Test'''
 |-
+|&nbsp; &nbsp; &nbsp; &nbsp; Then,
 |-
+|
 |-
+|
 |-
+|
 |}
@@ Line 27: / Line 42: @@
 !Step 1: &nbsp;
 |-
+|We begin by using the Ratio Test.
 |-
+|We have
 |-
+|
 |}
@@ Line 37: / Line 61: @@
 !Step 2: &nbsp;
 |-
+|Since
 |-
+|Therefore, the series converges.
 |}
@@ Line 47: / Line 75: @@
 !Step 1: &nbsp;
 |-
+|For
 |-
+|&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>
 |}
@@ Line 57: / Line 93: @@
 !Step 2: &nbsp;
 |-
+|Also,
 |-
+|by the Alternating Series Test.
 |}
@@ Line 66: / Line 106: @@
 !Final Answer: &nbsp;
 |-
-|&nbsp;&nbsp; '''(a)'''
+|&nbsp;&nbsp; '''(a)''' &nbsp;&nbsp; converges
 |-
-|&nbsp;&nbsp; '''(b)'''
+|&nbsp;&nbsp; '''(b)''' &nbsp;&nbsp; converges
 |}
 [[009C_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

@@ Line 1: / Line 1: @@
-<span class="exam">Compute
+<span class="exam"> Determine if the following series converges or diverges. Please give your reason(s).
-::<span class="exam">a) <math style="vertical-align: -12px">\lim_{n\rightarrow \infty} \frac{3-2n^2}{5n^2+n+1}</math>
+::<span class="exam">a) <math>\sum_{n=1}^{+\infty} \frac{n!}{(2n)!}</math>
-::<span class="exam">b) <math style="vertical-align: -12px">\lim_{n\rightarrow \infty} \frac{\ln n}{\ln 3n}</math>
+::<span class="exam">b) <math>\sum_{n=1}^{+\infty} (-1)^n\frac{1}{n+1}</math>
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"