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| − | <span class="exam">If <math>\sum_{n=0}^\infty c_nx^n</math> converges, does it follow that the following series converges? | + | <span class="exam">If <math>\sum_{n=0}^\infty c_nx^n</math> converges, does it follow that the following series converges? |
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| − | ::<span class="exam">a) <math>\sum_{n=0}^\infty c_n\bigg(\frac{x}{2}\bigg)^n</math>
| + | <span class="exam">(a) <math>\sum_{n=0}^\infty c_n\bigg(\frac{x}{2}\bigg)^n</math> |
| − | ::<span class="exam">b) <math>\sum_{n=0}^\infty c_n(-x)^n </math>
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| | + | <span class="exam">(b) <math>\sum_{n=0}^\infty c_n(-x)^n </math> |
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| + | <hr> |
| − | !Foundations:
| + | [[009C Sample Midterm 2, Problem 5 Solution|'''<u>Solution</u>''']] |
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| − | |A geometric series <math>\sum_{n=0}^{\infty} ar^n</math> converges if <math>|r|<1.</math> | |
| − | |}
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| − | '''Solution:''' | + | [[009C Sample Midterm 2, Problem 5 Detailed Solution|'''<u>Detailed Solution</u>''']] |
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| − | '''(a)'''
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 1:
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| − | |-
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| − | |First, we notice that <math>\sum_{n=0}^\infty c_nx^n</math> is a geometric series.
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| − | |-
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| − | |We have <math>r=x.</math>
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| − | |-
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| − | |Since this series converges,
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| − | |-
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| − | | <math>|r|=|x|<1.</math>
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| − | |}
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 2:
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| − | |-
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| − | |The series <math>\sum_{n=0} c_n\bigg(\frac{x}{2}\bigg)^n</math> is also a geometric series.
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| − | |-
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| − | |For this series, <math>r=\frac{x}{2}.</math>
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| − | |-
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| − | |Now, we notice
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| − | <math>\begin{array}{rcl}
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| − | \displaystyle{|r|} & = & \displaystyle{\bigg|\frac{x}{2}\bigg|}\\
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| − | &&\\
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| − | & = & \displaystyle{\frac{|x|}{2}}\\
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| − | &&\\
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| − | & < & \displaystyle{\frac{1}{2}}
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| − | \end{array}</math>
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| − | |-
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| − | |since <math>|x|<1.</math>
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| − | |-
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| − | | Since <math>|r|<1,</math> this series converges.
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| − | |}
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| − | '''(b)'''
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 1:
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| − | |-
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| − | |First, we notice that <math>\sum_{n=0}^\infty c_nx^n</math> is a geometric series.
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| − | |-
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| − | |We have <math>r=x.</math>
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| − | |-
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| − | |Since this series converges,
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| − | |-
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| − | | <math>|r|=|x|<1.</math>
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| − | |}
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| − |
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 2:
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| − | |-
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| − | |The series <math>\sum_{n=0}^\infty c_n(-x)^n</math> is also a geometric series.
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| − | |-
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| − | |For this series, <math>r=-x.</math>
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| − | |-
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| − | |Now, we notice
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| − | |-
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| − | <math>\begin{array}{rcl}
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| − | \displaystyle{|r|} & = & \displaystyle{|-x|}\\
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| − | &&\\
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| − | & = & \displaystyle{|x|}\\
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| − | &&\\
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| − | & < & \displaystyle{1}
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| − | \end{array}</math>
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| − | |-
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| − | |since <math>|x|<1.</math>
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| − | |-
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| − | |Since <math>|r|<1,</math> this series converges.
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| − | |}
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Final Answer:
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| − | |-
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| − | | '''(a)''' The series converges.
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| − | |-
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| − | | '''(b)''' The series converges.
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| − | |}
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| | [[009C_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']] | | [[009C_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']] |
