|
|
| (6 intermediate revisions by 2 users not shown) |
| Line 3: |
Line 3: |
| | <span class="exam">(a) <math style="vertical-align: -14px">4-2+1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdots</math> | | <span class="exam">(a) <math style="vertical-align: -14px">4-2+1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdots</math> |
| | | | |
| − | <span class="exam">(b) <math>\sum_{n=1}^{+\infty} \frac{1}{(2n-1)(2n+1)}</math> | + | <span class="exam">(b) <math>\sum_{n=1}^{\infty} \frac{1}{(2n-1)(2n+1)}</math> |
| | | | |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| + | <hr> |
| − | !Foundations:
| + | [[009C Sample Final 2, Problem 2 Solution|'''<u>Solution</u>''']] |
| − | |-
| |
| − | |'''1.''' The sum of a convergent geometric series is <math>\frac{a}{1-r}</math> | |
| − | |-
| |
| − | | where <math style="vertical-align: 0px">r</math> is the ratio of the geometric series
| |
| − | |-
| |
| − | | and <math style="vertical-align: 0px">a</math> is the first term of the series.
| |
| − | |-
| |
| − | |'''2.''' The <math style="vertical-align: 0px">n</math>th partial sum, <math style="vertical-align: -3px">s_n</math> for a series <math>\sum_{n=1}^\infty a_n </math> is defined as
| |
| − | |-
| |
| − | |
| |
| − | <math>s_n=\sum_{i=1}^n a_i.</math>
| |
| − | |}
| |
| | | | |
| | | | |
| − | '''Solution:''' | + | [[009C Sample Final 2, Problem 2 Detailed Solution|'''<u>Detailed Solution</u>''']] |
| | | | |
| − | '''(a)'''
| |
| | | | |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| |
| − | !Step 1:
| |
| − | |-
| |
| − | |
| |
| − | |-
| |
| − | |
| |
| − | |-
| |
| − | |
| |
| − | |}
| |
| − |
| |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| |
| − | !Step 2:
| |
| − | |-
| |
| − | |
| |
| − | |-
| |
| − | |
| |
| − | |}
| |
| − |
| |
| − | '''(b)'''
| |
| − |
| |
| − | {| 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)'''
| |
| − | |-
| |
| − | | '''(b)'''
| |
| − | |}
| |
| | [[009C_Sample_Final_2|'''<u>Return to Sample Exam</u>''']] | | [[009C_Sample_Final_2|'''<u>Return to Sample Exam</u>''']] |
Latest revision as of 18:22, 2 December 2017
For each of the following series, find the sum if it converges. If it diverges, explain why.
(a) 
(b) 
Solution
Detailed Solution
Return to Sample Exam