Difference between revisions of "009C Sample Midterm 1, Problem 4"
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|'''Direct Comparison Test''' | |'''Direct Comparison Test''' | ||
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| − | | Let <math>\{a_n\}</math> and <math>\{b_n\}</math> be positive sequences where <math style="vertical-align: -3px">a_n\le b_n</math> | + | | Let <math>\{a_n\}</math> and <math>\{b_n\}</math> be positive sequences where <math style="vertical-align: -3px">a_n\le b_n</math> |
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| − | | for all <math style="vertical-align: -3px">n\ge N</math> for some <math style="vertical-align: -3px">N\ge 1.</math> | + | | for all <math style="vertical-align: -3px">n\ge N</math> for some <math style="vertical-align: -3px">N\ge 1.</math> |
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| − | | '''1.''' If <math>\sum_{n=1}^\infty b_n</math> converges, then <math>\sum_{n=1}^\infty a_n</math> converges. | + | | '''1.''' If <math>\sum_{n=1}^\infty b_n</math> converges, then <math>\sum_{n=1}^\infty a_n</math> converges. |
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| − | | '''2.''' If <math>\sum_{n=1}^\infty a_n</math> diverges, then <math>\sum_{n=1}^\infty b_n</math> diverges. | + | | '''2.''' If <math>\sum_{n=1}^\infty a_n</math> diverges, then <math>\sum_{n=1}^\infty b_n</math> diverges. |
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| <math>\frac{1}{n^23^n}>0</math> | | <math>\frac{1}{n^23^n}>0</math> | ||
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| − | |for all <math style="vertical-align: -3px">n\ge 1.</math> | + | |for all <math style="vertical-align: -3px">n\ge 1.</math> |
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|This means that we can use a comparison test on this series. | |This means that we can use a comparison test on this series. | ||
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| − | |Let <math style="vertical-align: -14px">a_n=\frac{1}{n^23^n}.</math> | + | |Let <math style="vertical-align: -14px">a_n=\frac{1}{n^23^n}.</math> |
|} | |} | ||
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!Step 2: | !Step 2: | ||
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| − | |Let <math style="vertical-align: -14px">b_n=\frac{1}{n^2}.</math> | + | |Let <math style="vertical-align: -14px">b_n=\frac{1}{n^2}.</math> |
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|We want to compare the series in this problem with | |We want to compare the series in this problem with | ||
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| <math>\sum_{n=1}^\infty b_n=\sum_{n=1}^\infty \frac{1}{n^2}.</math> | | <math>\sum_{n=1}^\infty b_n=\sum_{n=1}^\infty \frac{1}{n^2}.</math> | ||
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| − | |This is a <math style="vertical-align: -4px">p</math>-series with <math style="vertical-align: -4px">p=2.</math> | + | |This is a <math style="vertical-align: -4px">p</math>-series with <math style="vertical-align: -4px">p=2.</math> |
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| − | |Hence, <math>\sum_{n=1}^\infty b_n</math> converges. | + | |Hence, <math>\sum_{n=1}^\infty b_n</math> converges. |
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!Step 3: | !Step 3: | ||
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| − | |Also, we have <math style="vertical-align: -4px">a_n<b_n</math> since | + | |Also, we have <math style="vertical-align: -4px">a_n<b_n</math> since |
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| <math>\frac{1}{n^23^n}<\frac{1}{n^2}</math> | | <math>\frac{1}{n^23^n}<\frac{1}{n^2}</math> | ||
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| − | |for all <math style="vertical-align: -3px">n\ge 1.</math> | + | |for all <math style="vertical-align: -3px">n\ge 1.</math> |
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| − | |Therefore, the series <math>\sum_{n=1}^\infty a_n</math> converges | + | |Therefore, the series <math>\sum_{n=1}^\infty a_n</math> converges |
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|by the Direct Comparison Test. | |by the Direct Comparison Test. | ||
Revision as of 17:55, 26 February 2017
Determine the convergence or divergence of the following series.
Be sure to justify your answers!
| Foundations: |
|---|
| Direct Comparison Test |
| Let and 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 \{b_n\}} be positive sequences where 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 a_n\le b_n} |
| for all 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 n\ge N} for some 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 N\ge 1.} |
| 1. If 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 b_n} converges, then 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 a_n} converges. |
| 2. If 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 a_n} diverges, then 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 b_n} diverges. |
Solution:
| Step 1: |
|---|
| First, we note that |
| 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 \frac{1}{n^23^n}>0} |
| for all 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 n\ge 1.} |
| This means that we can use a comparison test on this series. |
| Let 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 a_n=\frac{1}{n^23^n}.} |
| Step 2: |
|---|
| Let 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 b_n=\frac{1}{n^2}.} |
| We want to compare the series in this problem with |
| 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 b_n=\sum_{n=1}^\infty \frac{1}{n^2}.} |
| This is a 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 p} -series with 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 p=2.} |
| Hence, 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 b_n} converges. |
| Step 3: |
|---|
| Also, we have 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 a_n<b_n} since |
| 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 \frac{1}{n^23^n}<\frac{1}{n^2}} |
| for all 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 n\ge 1.} |
| Therefore, the series 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 a_n} converges |
| by the Direct Comparison Test. |
| Final Answer: |
|---|
| converges |