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| | + | [[009C Sample Midterm 2, Problem 5 Solution|'''<u>Solution</u>''']] |
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Foundations:
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| − | |-
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| − | |If a power series converges, then it has a nonempty interval of convergence.
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| − | |}
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| | + | [[009C Sample Midterm 2, Problem 5 Detailed Solution|'''<u>Detailed Solution</u>''']] |
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| − | '''Solution:'''
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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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| − | |Assume that the power series <math style="vertical-align: -19px">\sum_{n=0}^\infty c_nx^n</math> converges.
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| − | |-
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| − | |Let <math style="vertical-align: 0px">R</math> be the radius of convergence of this power series.
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| − | |-
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| − | |So, the power series
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| − | | <math style="vertical-align: -19px">\sum_{n=0}^\infty c_nx^n</math>
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| − | |-
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| − | |converges in the interval <math style="vertical-align: -5px">(-R,R).</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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| − | |Let <math style="vertical-align: -5px">a\in (-2R,2R).</math> Then, <math style="vertical-align: -13px">\frac{a}{2} \in (-R,R).</math>
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| − | |So,
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| − | | <math>\sum_{n=0}^\infty c_n\bigg(\frac{a}{2}\bigg)^n</math>
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| − | |converges by assumption.
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| − | |Since <math style="vertical-align: 0px">a</math> was an arbitrary number in the interval <math style="vertical-align: -5px">a\in (-2R,2R),</math>
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| − | |-
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| − | | <math>\sum_{n=0}^\infty c_n\bigg(\frac{x}{2}\bigg)^n</math>
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| − | |-
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| − | |converges in the interval <math style="vertical-align: -5px">(-2R,2R).</math>
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| − | |}
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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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| − |
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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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| − | |
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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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| − | | '''(a)''' converges
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| − | | '''(b)''' 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>''']] |
If 
converges, does it follow that the following series converges?
(a) 
(b) 
Solution
Detailed Solution
Return to Sample Exam