009C Sample Final 2, Problem 7 - Revision history

Kayla Murray: Replaced content with "(a) Consider the function $f(x)=\bigg(1-\frac{1}{2}x\bigg)^{-2}.$ Find the first three terms of its..."

2017-12-03T01:33:03Z

Replaced content with "<span class="exam">(a) Consider the function <math style="vertical-align: -16px">f(x)=\bigg(1-\frac{1}{2}x\bigg)^{-2}.</math> Find the first three terms of its..."

Grad at 16:37, 12 March 2017

2017-03-12T16:37:53Z

Grad at 00:02, 11 March 2017

2017-03-11T00:02:29Z

Kayla Murray at 18:50, 5 March 2017

2017-03-05T18:50:49Z

Kayla Murray at 18:39, 5 March 2017

2017-03-05T18:39:13Z

Kayla Murray at 18:29, 5 March 2017

2017-03-05T18:29:34Z

Kayla Murray at 01:15, 5 March 2017

2017-03-05T01:15:19Z

Kayla Murray at 00:49, 5 March 2017

2017-03-05T00:49:09Z

Kayla Murray at 00:43, 5 March 2017

2017-03-05T00:43:44Z

Kayla Murray at 18:23, 18 February 2017

2017-02-18T18:23:28Z

@@ Line 3: / Line 3: @@
 <span class="exam">(b) Find its radius of convergence.
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+<hr>
-!Foundations: &nbsp;
+[[009C Sample Final 2, Problem 7 Solution|'''<u>Solution</u>''']]
-'''Solution:'''
+[[009C Sample Final 2, Problem 7 Detailed Solution|'''<u>Detailed Solution</u>''']]
 [[009C_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]

@@ Line 83: / Line 83: @@
 !Step 1: &nbsp;
 |-
-|The Maclaurin series of &nbsp;<math>\frac{1}{(1-x)^2}</math>&nbsp; is
+|By taking the derivative of the known series
 |-
 |&nbsp; &nbsp; &nbsp; &nbsp;<math>\sum_{n=0}^\infty (n+1)x^n.</math>
 |-
-|So, the Maclaurin series of &nbsp;<math>\frac{1}{(1-\frac{1}{2}x)^2}</math>&nbsp; is
+|Letting <math style="vertical-align: -5px"> x/2 </math> &thinsp; play the role of <math style="vertical-align: -4px">x,</math> the Maclaurin series of &nbsp;<math>\frac{1}{(1-\frac{1}{2}x)^2}</math>&nbsp; is
 |-
 |&nbsp; &nbsp; &nbsp; &nbsp;<math>\sum_{n=0}^\infty (n+1)\bigg(\frac{1}{2}x\bigg)^n=\sum_{n=0}^\infty \frac{(n+1)x^n}{2^n}.</math>

@@ Line 40: / Line 40: @@
 |-
+|
-<table border="1" cellspacing="0" cellpadding="6" align = "center">
+<table border="1" cellspacing="0" cellpadding="11" align = "center">
    <tr>
      <td align = "center"><math> n</math></td>

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 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Foundations: &nbsp;
 |-
+|
 |-
+|&nbsp; &nbsp; &nbsp; &nbsp; Then,
 |-
+|
 |-
+|
 |-
+|
 |}
@@ Line 25: / Line 35: @@
 !Step 1: &nbsp;
 |-
+|We begin by finding the coefficients of the Maclaurin series for &nbsp;<math>f(x)=\frac{1}{(1-\frac{1}{2}x)^2}.</math>
 |-
+|
 |-
+|
@@ Line 35: / Line 73: @@
 !Step 2: &nbsp;
 |-
+|So, the first three terms of the Binomial Series is
 |-
+|&nbsp; &nbsp; &nbsp; &nbsp; <math>1+x+\frac{3}{4}x^2.</math>
 |}
@@ Line 82: / Line 120: @@
 !Final Answer: &nbsp;
 |-
-|&nbsp; &nbsp;'''(a)'''&nbsp; &nbsp;
+|&nbsp; &nbsp;'''(a)'''&nbsp; &nbsp; <math>1+x+\frac{3}{4}x^2</math>
 |-
 |&nbsp; &nbsp;'''(b)'''&nbsp; &nbsp; The radius of convergence is &nbsp;<math style="vertical-align: 0px">R=2.</math>
 |}
 [[009C_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]

@@ Line 59: / Line 59: @@
 !Step 2: &nbsp;
 |-
+|Now, we use the Ratio Test to determine the radius of convergence of this power series.
 |-
+|We have
 |}
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 !Final Answer: &nbsp;
 |-
-|&nbsp;&nbsp; '''(a)'''
+|&nbsp; &nbsp;'''(a)'''&nbsp; &nbsp;
 |-
-|&nbsp;&nbsp; '''(b)'''
+|&nbsp; &nbsp;'''(b)'''&nbsp; &nbsp; The radius of convergence is &nbsp;<math style="vertical-align: 0px">R=2.</math>
 |}
 [[009C_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]

← Older revision		Revision as of 01:15, 5 March 2017
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−	<span class="exam">(a) Consider the function  <math>f(x)=\bigg(1-\frac{1}{2}x\bigg)^{-2}.</math>  Find the first three terms of its Binomial Series.	+	<span class="exam">(a) Consider the function  <math style="vertical-align: -16px">f(x)=\bigg(1-\frac{1}{2}x\bigg)^{-2}.</math>  Find the first three terms of its Binomial Series.

	<span class="exam">(b) Find its radius of convergence.		<span class="exam">(b) Find its radius of convergence.

@@ Line 1: / Line 1: @@
-::<span class="exam">a) Consider the function <math>f(x)=\bigg(1-\frac{1}{2}x\bigg)^{-2}.</math> Find the first three terms of its Binomial Series.
+<span class="exam">(a) Consider the function &nbsp;<math>f(x)=\bigg(1-\frac{1}{2}x\bigg)^{-2}.</math>&nbsp; Find the first three terms of its Binomial Series.
-::<span class="exam">b) Find its radius of convergence.
+<span class="exam">(b) Find its radius of convergence.
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"