|
|
| Line 6: |
Line 6: |
| | | | |
| | | | |
| | + | <hr> |
| | + | (insert picture of handwritten solution) |
| | | | |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| + | [[009C Sample Midterm 2, Problem 5 Detailed Solution|'''<u>Detailed Solution for this Problem</u>''']] |
| − | !Foundations:
| |
| − | |-
| |
| − | |A geometric series <math>\sum_{n=0}^{\infty} ar^n</math> converges if <math style="vertical-align: -6px">|r|<1.</math>
| |
| − | |}
| |
| | | | |
| − |
| |
| − | '''Solution:'''
| |
| − |
| |
| − | '''(a)'''
| |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| |
| − | !Step 1:
| |
| − | |-
| |
| − | |First, we notice that <math>\sum_{n=0}^\infty c_nx^n</math> is a geometric series.
| |
| − | |-
| |
| − | |We have <math style="vertical-align: -1px">r=x.</math>
| |
| − | |-
| |
| − | |Since this series converges,
| |
| − | |-
| |
| − | | <math>|r|=|x|<1.</math>
| |
| − | |}
| |
| − |
| |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| |
| − | !Step 2:
| |
| − | |-
| |
| − | |The series <math>\sum_{n=0} c_n\bigg(\frac{x}{2}\bigg)^n</math> is also a geometric series.
| |
| − | |-
| |
| − | |For this series, <math style="vertical-align: -13px">r=\frac{x}{2}.</math>
| |
| − | |-
| |
| − | |Now, we notice
| |
| − | |-
| |
| − | |
| |
| − | <math>\begin{array}{rcl}
| |
| − | \displaystyle{|r|} & = & \displaystyle{\bigg|\frac{x}{2}\bigg|}\\
| |
| − | &&\\
| |
| − | & = & \displaystyle{\frac{|x|}{2}}\\
| |
| − | &&\\
| |
| − | & < & \displaystyle{\frac{1}{2}}
| |
| − | \end{array}</math>
| |
| − | |-
| |
| − | |since <math style="vertical-align: -5px">|x|<1.</math>
| |
| − | |-
| |
| − | | Since <math style="vertical-align: -5px">|r|<1,</math> this series converges.
| |
| − | |}
| |
| − |
| |
| − | '''(b)'''
| |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| |
| − | !Step 1:
| |
| − | |-
| |
| − | |First, we notice that <math>\sum_{n=0}^\infty c_nx^n</math> is a geometric series.
| |
| − | |-
| |
| − | |We have <math style="vertical-align: -1px">r=x.</math>
| |
| − | |-
| |
| − | |Since this series converges,
| |
| − | |-
| |
| − | | <math>|r|=|x|<1.</math>
| |
| − | |}
| |
| − |
| |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| |
| − | !Step 2:
| |
| − | |-
| |
| − | |The series <math>\sum_{n=0}^\infty c_n(-x)^n</math> is also a geometric series.
| |
| − | |-
| |
| − | |For this series, <math style="vertical-align: -1px">r=-x.</math>
| |
| − | |-
| |
| − | |Now, we notice
| |
| − | |-
| |
| − | |
| |
| − | <math>\begin{array}{rcl}
| |
| − | \displaystyle{|r|} & = & \displaystyle{|-x|}\\
| |
| − | &&\\
| |
| − | & = & \displaystyle{|x|}\\
| |
| − | &&\\
| |
| − | & < & \displaystyle{1}
| |
| − | \end{array}</math>
| |
| − | |-
| |
| − | |since <math style="vertical-align: -5px">|x|<1.</math>
| |
| − | |-
| |
| − | |Since <math style="vertical-align: -5px">|r|<1,</math> this series converges.
| |
| − | |}
| |
| − |
| |
| − |
| |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| |
| − | !Final Answer:
| |
| − | |-
| |
| − | | '''(a)''' converges (by the geometric series test)
| |
| − | |-
| |
| − | | '''(b)''' converges (by the geometric series test)
| |
| − | |}
| |
| | [[009C_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']] | | [[009C_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']] |
If Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \sum _{n=0}^{\infty }c_{n}x^{n}}
converges, does it follow that the following series converges?
(a) Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \sum _{n=0}^{\infty }c_{n}{\bigg (}{\frac {x}{2}}{\bigg )}^{n}}
(b) 
(insert picture of handwritten solution)
Detailed Solution for this Problem
Return to Sample Exam