009C Sample Final 2, Problem 4 - Revision history

Kayla Murray: Replaced content with "(a) Find the radius of convergence for the power series :: $\sum_{n=1}^{\infty} (-1)^n \frac{x^n}{n}.$ (b) Find the inter..."

2017-12-03T01:27:31Z

Replaced content with "<span class="exam">(a) Find the radius of convergence for the power series ::<math>\sum_{n=1}^{\infty} (-1)^n \frac{x^n}{n}.</math> <span class="exam">(b) Find the inter..."

Kayla Murray at 18:19, 17 March 2017

2017-03-17T18:19:04Z

Grad at 16:56, 12 March 2017

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Grad at 00:20, 11 March 2017

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Kayla Murray at 04:05, 5 March 2017

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Kayla Murray at 04:01, 5 March 2017

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Kayla Murray at 03:59, 5 March 2017

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Kayla Murray at 03:52, 5 March 2017

2017-03-05T03:52:47Z

Kayla Murray at 00:47, 5 March 2017

2017-03-05T00:47:46Z

@@ Line 5: / Line 5: @@
 <span class="exam">(b) Find the interval of convergence of the above series.
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+<hr>
-!Foundations: &nbsp;
+[[009C Sample Final 2, Problem 4 Solution|'''<u>Solution</u>''']]
-'''Solution:'''
+[[009C Sample Final 2, Problem 4 Detailed Solution|'''<u>Detailed Solution</u>''']]
 [[009C_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]

@@ Line 75: / Line 75: @@
 |First, let &nbsp;<math style="vertical-align: -1px">x=1.</math>
 |-
-|Then, the series becomes &nbsp;<math>\sum_{n=0}^\infty (-1)^n \frac{1}{n}.</math>
+|Then, the series becomes &nbsp;<math>\sum_{n=1}^\infty (-1)^n \frac{1}{n}.</math>
 |-
 |This is an alternating series.
 |-
 |Let &nbsp;<math style="vertical-align: -15px">b_n=\frac{1}{n}.</math>.
 |-
 |The sequence &nbsp;<math>\{b_n\}</math>&nbsp; is decreasing since
@@ Line 101: / Line 107: @@
 |Now, let &nbsp;<math style="vertical-align: -1px">x=-1.</math>
 |-
-|Then, the series becomes &nbsp;<math>\sum_{n=0}^\infty \frac{1}{n}.</math>
+|Then, the series becomes &nbsp;<math>\sum_{n=1}^\infty \frac{1}{n}.</math>
 |-
 |This is a &nbsp;<math>p</math>-series with &nbsp;<math>p=1.</math>&nbsp; Hence, the series diverges.

@@ Line 1: / Line 1: @@
 <span class="exam">(a) Find the radius of convergence for the power series
-::<math>\sum_{n=1}^{+\infty} (-1)^n \frac{x^n}{n}.</math>
+::<math>\sum_{n=1}^{\infty} (-1)^n \frac{x^n}{n}.</math>
 <span class="exam">(b) Find the interval of convergence of the above series.

@@ Line 38: / Line 38: @@
+|
 &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{(-1)^{n+1}(x)^{n+1}}{(n+1)}\frac{n}{(-1)^n(x)^n}\bigg|}\\
+\displaystyle{\lim_{n\rightarrow \infty}\bigg|\frac{a_{n+1}}{a_n}\bigg|} & = & \displaystyle{\lim_{n\rightarrow \infty} \bigg|\frac{(-1)^{n+1}(x)^{n+1}}{(n+1)}\cdot\frac{n}{(-1)^n(x)^n}\bigg|}\\
 &&\\
 & = & \displaystyle{\lim_{n\rightarrow \infty} \bigg|(-1)(x)\frac{n}{n+1}\bigg|}\\

@@ Line 94: / Line 94: @@
 |-
 |and we include &nbsp;<math style="vertical-align: -1px">x=1</math>&nbsp; in our interval.
 |}
@@ Line 102: / Line 120: @@
 |&nbsp; &nbsp; '''(a)''' &nbsp; &nbsp; The radius of convergence is &nbsp;<math style="vertical-align: -1px">R=1.</math>
 |-
-|&nbsp; &nbsp; '''(b)'''
+|&nbsp; &nbsp; '''(b)''' &nbsp; &nbsp; <math>(-1,1]</math>
 |}
 [[009C_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]

@@ Line 63: / Line 63: @@
 !Step 1: &nbsp;
 |-
+|First, note that &nbsp;<math style="vertical-align: -5px">|x|<1</math>&nbsp; corresponds to the interval &nbsp;<math style="vertical-align: -4px">(-1,1).</math>
 |-
+|To obtain the interval of convergence, we need to test the endpoints of this interval
 |-
+|for convergence since the Ratio Test is inconclusive when &nbsp;<math style="vertical-align: -1px">R=1.</math>
 |}

@@ Line 32: / Line 32: @@
 !Step 1: &nbsp;
 |-
+|We use the Ratio Test to determine the radius of convergence.
 |-
+|We have
 |-
+|
 |}
@@ Line 42: / Line 53: @@
 !Step 2: &nbsp;
 |-
+|The Ratio Test tells us this series is absolutely convergent if &nbsp;<math style="vertical-align: -5px">|x|<1.</math>
 |-
+|Hence, the Radius of Convergence of this series is &nbsp;<math style="vertical-align: -1px">R=1.</math>
 |}
@@ Line 71: / Line 82: @@
 !Final Answer: &nbsp;
 |-
-|&nbsp;&nbsp; '''(a)'''
+|&nbsp; &nbsp; '''(a)''' &nbsp; &nbsp; The radius of convergence is &nbsp;<math style="vertical-align: -1px">R=1.</math>
 |-
-|&nbsp;&nbsp; '''(b)'''
+|&nbsp; &nbsp; '''(b)'''
 |}
 [[009C_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]

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