009C Sample Final 2, Problem 1 - Revision history

Kayla Murray at 01:20, 3 December 2017

2017-12-03T01:20:47Z

2017-03-18T19:15:07Z

2017-03-05T01:43:06Z

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@@ Line 5: / Line 5: @@
 <span class="exam">(b) &nbsp;<math style="vertical-align: -15px">a_n=\bigg(\frac{n}{n+1}\bigg)^n</math>
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+<hr>
-!Foundations: &nbsp;
+[[009C Sample Final 2, Problem 1 Solution|'''<u>Solution</u>''']]
-'''Solution:'''
+[[009C Sample Final 2, Problem 1 Detailed Solution|'''<u>Detailed Solution</u>''']]
 [[009C_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]

@@ Line 41: / Line 41: @@
 |-
 |&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
-\displaystyle{\lim_{n\rightarrow \infty}\frac{\ln(n)}{\ln(n+1)}} & \overset{L'H}{=} & \displaystyle{\lim_{x\rightarrow \infty} \frac{\frac{1}{x}}{\frac{1}{x+1}}}\\
+\displaystyle{\lim_{n\rightarrow \infty}\frac{\ln(n)}{\ln(n+1)}} & \overset{L'H}{=} & \displaystyle{\lim_{x\rightarrow \infty} \frac{\big(\frac{1}{x}\big)}{\big(\frac{1}{x+1}\big)}}\\
 &&\\
 & = & \displaystyle{\lim_{x\rightarrow \infty} \frac{x+1}{x}}\\

@@ Line 11: / Line 11: @@
 |-
+|
-&nbsp; &nbsp; &nbsp; &nbsp; Suppose that <math>\lim_{x\rightarrow \infty} f(x)</math> &nbsp; and <math>\lim_{x\rightarrow \infty} g(x)</math> &nbsp; are both zero or both &nbsp; <math style="vertical-align: -1px">\pm \infty .</math>
+&nbsp; &nbsp; &nbsp; &nbsp; Suppose that &nbsp;<math>\lim_{x\rightarrow \infty} f(x)</math> &nbsp; and &nbsp;<math>\lim_{x\rightarrow \infty} g(x)</math> &nbsp; are both zero or both &nbsp; <math style="vertical-align: -1px">\pm \infty .</math>
 |-
+|
-&nbsp; &nbsp; &nbsp; &nbsp;If <math>\lim_{x\rightarrow \infty} \frac{f'(x)}{g'(x)}</math> &nbsp; is finite or &nbsp; <math style="vertical-align: -4px">\pm \infty ,</math>
+&nbsp; &nbsp; &nbsp; &nbsp;If &nbsp;<math>\lim_{x\rightarrow \infty} \frac{f'(x)}{g'(x)}</math> &nbsp; is finite or &nbsp; <math style="vertical-align: -4px">\pm \infty ,</math>
 |-
+|
-&nbsp; &nbsp; &nbsp; &nbsp;then <math>\lim_{x\rightarrow \infty} \frac{f(x)}{g(x)}=\lim_{x\rightarrow \infty} \frac{f'(x)}{g'(x)}.</math>
+&nbsp; &nbsp; &nbsp; &nbsp;then &nbsp;<math>\lim_{x\rightarrow \infty} \frac{f(x)}{g(x)}=\lim_{x\rightarrow \infty} \frac{f'(x)}{g'(x)}.</math>
 |}
@@ Line 109: / Line 109: @@
 !Step 4: &nbsp;
 |-
-|Since &nbsp;<math>\ln y= -1,</math> we know
+|Since &nbsp;<math>\ln y= -1,</math>&nbsp; we know
 |-
 |&nbsp; &nbsp; &nbsp; &nbsp; <math>y=e^{-1}.</math>

@@ Line 56: / Line 56: @@
 !Step 1: &nbsp;
 |-
+|Let
 |-
+|We then take the natural log of both sides to get
 |-
+|&nbsp; &nbsp; &nbsp; &nbsp; <math>\ln y = \ln\bigg(\lim_{n\rightarrow \infty} \bigg(\frac{n}{n+1}\bigg)^n\bigg).</math>
 |}
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Step 2: &nbsp;
 |-
+|
 |-
+|
 |}
@@ Line 77: / Line 120: @@
 |&nbsp;&nbsp; '''(a)''' &nbsp; &nbsp; <math>1</math>
 |-
-|&nbsp;&nbsp; '''(b)'''
+|&nbsp;&nbsp; '''(b)''' &nbsp; &nbsp; <math>e^{-1}</math>
 |}
 [[009C_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]

@@ Line 28: / Line 28: @@
 !Step 1: &nbsp;
 |-
+|First, we notice that &nbsp;<math>\lim_{n\rightarrow \infty} \frac{\ln(n)}{\ln(n+1)}</math>&nbsp; has the form &nbsp;<math>\frac{\infty}{\infty}.</math>
 |-
+|So, we can use L'Hopital's Rule. To begin, we write
 |-
+|&nbsp; &nbsp; &nbsp; &nbsp;<math>\lim_{n\rightarrow \infty}\frac{\ln(n)}{\ln(n+1)}=\lim_{x\rightarrow \infty} \frac{\ln(x)}{\ln(x+1)}.</math>
 |}
@@ Line 38: / Line 38: @@
 !Step 2: &nbsp;
 |-
+|Now, using L'Hopital's rule, we get
 |-
+|&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
 |}
@@ Line 67: / Line 75: @@
 !Final Answer: &nbsp;
 |-
-|&nbsp;&nbsp; '''(a)'''
+|&nbsp;&nbsp; '''(a)''' &nbsp; &nbsp; <math>1</math>
 |-
 |&nbsp;&nbsp; '''(b)'''
 |}
 [[009C_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]

@@ Line 1: / Line 1: @@
 <span class="exam">Test if the following sequences converge or diverge. Also find the limit of each convergent sequence.
-<span class="exam">(a) &nbsp;<math style="vertical-align: -12px">a_n=\frac{\ln(n)}{\ln(n+1)}</math>
+<span class="exam">(a) &nbsp;<math style="vertical-align: -16px">a_n=\frac{\ln(n)}{\ln(n+1)}</math>
-<span class="exam">(b) &nbsp;<math style="vertical-align: -12px">a_n=\bigg(\frac{n}{n+1}\bigg)^n</math>
+<span class="exam">(b) &nbsp;<math style="vertical-align: -15px">a_n=\bigg(\frac{n}{n+1}\bigg)^n</math>
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"

@@ Line 8: / Line 8: @@
 !Foundations: &nbsp;
 |-
+|'''L'Hopital's Rule'''
 |-
+|
 |-
+|
 |-
+|
 |}

@@ Line 1: / Line 1: @@
-<span class="exam">Compute
+<span class="exam">Test if the following sequences converge or diverge. Also find the limit of each convergent sequence.
-::<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 style="vertical-align: -12px">a_n=\frac{\ln(n)}{\ln(n+1)}</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 style="vertical-align: -12px">a_n=\bigg(\frac{n}{n+1}\bigg)^n</math>
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"