Difference between revisions of "009C Sample Midterm 1, Problem 4"

Revision as of 17:05, 4 November 2017

Determine the convergence or divergence of the following series.

Be sure to justify your answers!

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=1}^\infty \frac{1}{n^23^n}}

(insert picture of handwritten solution)

Detailed Solution for this Problem

Return to Sample Exam

@@ Line 6: / Line 6: @@
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+<hr>
-!Foundations: &nbsp;
+(insert picture of handwritten solution)
-|-
-|'''Direct Comparison Test'''
-|-
-|&nbsp; &nbsp; &nbsp; &nbsp; Let &nbsp;<math>\{a_n\}</math>&nbsp; and &nbsp;<math>\{b_n\}</math>&nbsp; be positive sequences where &nbsp;<math style="vertical-align: -3px">a_n\le b_n</math>
-|-
-|&nbsp; &nbsp; &nbsp; &nbsp; for all &nbsp;<math style="vertical-align: -3px">n\ge N</math>&nbsp; for some &nbsp;<math style="vertical-align: -3px">N\ge 1.</math>
-|-
-|&nbsp; &nbsp; &nbsp; &nbsp; '''1.''' If &nbsp;<math>\sum_{n=1}^\infty b_n</math>&nbsp; converges, then &nbsp;<math>\sum_{n=1}^\infty a_n</math>&nbsp; converges.
-|-
-|&nbsp; &nbsp; &nbsp; &nbsp; '''2.''' If &nbsp;<math>\sum_{n=1}^\infty a_n</math>&nbsp; diverges, then &nbsp;<math>\sum_{n=1}^\infty b_n</math>&nbsp; diverges.
-|}
+[[009C Sample Midterm 1, Problem 4 Detailed Solution|'''<u>Detailed Solution for this Problem</u>''']]
-'''Solution:'''
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Step 1: &nbsp;
-|-
-|First, we note that
-|-
-|&nbsp; &nbsp; &nbsp; &nbsp; <math>\frac{1}{n^23^n}>0</math>
-|-
-|for all &nbsp;<math style="vertical-align: -3px">n\ge 1.</math>
-|-
-|This means that we can use a comparison test on this series.
-|-
-|Let &nbsp;<math style="vertical-align: -14px">a_n=\frac{1}{n^23^n}.</math>
-|}
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Step 2: &nbsp;
-|-
-|Let &nbsp;<math style="vertical-align: -14px">b_n=\frac{1}{n^2}.</math>
-|-
-|We want to compare the series in this problem with
-|-
-|&nbsp; &nbsp; &nbsp; &nbsp; <math>\sum_{n=1}^\infty b_n=\sum_{n=1}^\infty \frac{1}{n^2}.</math>
-|-
-|This is a &nbsp;<math style="vertical-align: -4px">p</math>-series with &nbsp;<math style="vertical-align: -4px">p=2.</math>
-|-
-|Hence, &nbsp;<math>\sum_{n=1}^\infty b_n</math>&nbsp; converges.
-|}
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Step 3: &nbsp;
-|-
-|Also, we have &nbsp;<math style="vertical-align: -4px">a_n<b_n</math>&nbsp; since
-|-
-|&nbsp; &nbsp; &nbsp; &nbsp; <math>\frac{1}{n^23^n}<\frac{1}{n^2}</math>
-|-
-|for all &nbsp;<math style="vertical-align: -3px">n\ge 1.</math>
-|-
-|Therefore, the series &nbsp;<math>\sum_{n=1}^\infty a_n</math>&nbsp; converges
-|-
-|by the Direct Comparison Test.
-|}
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Final Answer: &nbsp;
-|-
-|&nbsp; &nbsp; &nbsp; &nbsp; converges (by the Direct Comparison Test)
-|-
-|
-|}
 [[009C_Sample_Midterm_1|'''<u>Return to Sample Exam</u>''']]

Difference between revisions of "009C Sample Midterm 1, Problem 4"

Revision as of 17:05, 4 November 2017

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