|
|
| (11 intermediate revisions by the same user not shown) |
| Line 1: |
Line 1: |
| | <span class="exam"> Consider the power series | | <span class="exam"> Consider the power series |
| | | | |
| − | ::<math>\sum_{n=0}^\infty (-1)^n \frac{x^{n+1}}{n+1}</math> | + | ::<math>\sum_{n=0}^\infty (-1)^n \frac{x^{n+1}}{n+1}.</math> |
| | | | |
| | <span class="exam">(a) Find the radius of convergence of the above power series. | | <span class="exam">(a) Find the radius of convergence of the above power series. |
| Line 13: |
Line 13: |
| | ::<math>\sum_{n=0}^\infty \frac{1}{(n+1)3^{n+1}}</math> | | ::<math>\sum_{n=0}^\infty \frac{1}{(n+1)3^{n+1}}</math> |
| | | | |
| − | <span class="exam">converge? If so, find its sum. | + | <span class="exam">converge? |
| | | | |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| + | <hr> |
| − | !Foundations:
| + | [[009C Sample Final 3, Problem 6 Solution|'''<u>Solution</u>''']] |
| − | |- | |
| − | |'''Ratio Test'''
| |
| − | |-
| |
| − | | Let <math style="vertical-align: -7px">\sum a_n</math> be a series and <math>L=\lim_{n\rightarrow \infty}\bigg|\frac{a_{n+1}}{a_n}\bigg|.</math>
| |
| − | |-
| |
| − | | Then,
| |
| − | |-
| |
| − | |
| |
| − | If <math style="vertical-align: -4px">L<1,</math> the series is absolutely convergent.
| |
| − | |-
| |
| − | |
| |
| − | If <math style="vertical-align: -4px">L>1,</math> the series is divergent.
| |
| − | |-
| |
| − | |
| |
| − | If <math style="vertical-align: -4px">L=1,</math> the test is inconclusive.
| |
| − | |}
| |
| | | | |
| | | | |
| − | '''Solution:''' | + | [[009C Sample Final 3, Problem 6 Detailed Solution|'''<u>Detailed Solution</u>''']] |
| | | | |
| − | '''(a)'''
| |
| | | | |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| |
| − | !Step 1:
| |
| − | |-
| |
| − | |We use the Ratio Test to determine the radius of convergence.
| |
| − | |-
| |
| − | |We have
| |
| − | |-
| |
| − | |
| |
| − | <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+2}}{(n+2)}\frac{n+1}{(-1)^n(x)^{n+1}}\bigg|}\\
| |
| − | &&\\
| |
| − | & = & \displaystyle{\lim_{n\rightarrow \infty} \bigg|(-1)(x)\frac{n+1}{n+2}\bigg|}\\
| |
| − | &&\\
| |
| − | & = & \displaystyle{\lim_{n\rightarrow \infty} |x|\frac{n+1}{n+2}}\\
| |
| − | &&\\
| |
| − | & = & \displaystyle{|x|\lim_{n\rightarrow \infty} \frac{n+1}{n+2}}\\
| |
| − | &&\\
| |
| − | & = & \displaystyle{|x|.}
| |
| − | \end{array}</math>
| |
| − | |}
| |
| − |
| |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| |
| − | !Step 2:
| |
| − | |-
| |
| − | |The Ratio Test tells us this series is absolutely convergent if <math style="vertical-align: -5px">|x|<1.</math>
| |
| − | |-
| |
| − | |Hence, the Radius of Convergence of this series is <math style="vertical-align: -1px">R=1.</math>
| |
| − | |}
| |
| − |
| |
| − | '''(b)'''
| |
| − |
| |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| |
| − | !Step 1:
| |
| − | |-
| |
| − | |
| |
| − | |-
| |
| − | |
| |
| − | |-
| |
| − | |
| |
| − | |}
| |
| − |
| |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| |
| − | !Step 2:
| |
| − | |-
| |
| − | |
| |
| − | |-
| |
| − | |
| |
| − | |}
| |
| − |
| |
| − | '''(c)'''
| |
| − |
| |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| |
| − | !Step 1:
| |
| − | |-
| |
| − | |
| |
| − | |-
| |
| − | |
| |
| − | |-
| |
| − | |
| |
| − | |}
| |
| − |
| |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| |
| − | !Step 2:
| |
| − | |-
| |
| − | |
| |
| − | |-
| |
| − | |
| |
| − | |}
| |
| − |
| |
| − | '''(d)'''
| |
| − |
| |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| |
| − | !Step 1:
| |
| − | |-
| |
| − | |
| |
| − | |-
| |
| − | |
| |
| − | |-
| |
| − | |
| |
| − | |}
| |
| − |
| |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| |
| − | !Step 2:
| |
| − | |-
| |
| − | |
| |
| − | |-
| |
| − | |
| |
| − | |}
| |
| − |
| |
| − |
| |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| |
| − | !Final Answer:
| |
| − | |-
| |
| − | | '''(a)''' The radius of convergence is <math style="vertical-align: -1px">R=1.</math>
| |
| − | |-
| |
| − | | '''(b)'''
| |
| − | |-
| |
| − | | '''(c)'''
| |
| − | |-
| |
| − | | '''(d)'''
| |
| − | |}
| |
| | [[009C_Sample_Final_3|'''<u>Return to Sample Exam</u>''']] | | [[009C_Sample_Final_3|'''<u>Return to Sample Exam</u>''']] |
