Difference between revisions of "009C Sample Final 3, Problem 3"
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| Let <math>\{a_n\}</math> and <math>\{b_n\}</math> be positive sequences. | | Let <math>\{a_n\}</math> and <math>\{b_n\}</math> be positive sequences. | ||
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| − | | If <math>\lim_{n\rightarrow \infty} \frac{a_n}{b_n}=L,</math> where <math>L</math> is a positive real number, | + | | If <math style="vertical-align: -16px">\lim_{n\rightarrow \infty} \frac{a_n}{b_n}=L,</math> where <math style="vertical-align: -1px">L</math> is a positive real number, |
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| − | | then <math>\sum_{n=1}^\infty a_n</math> and <math>\sum_{n=1}^\infty b_n</math> either both converge or both diverge. | + | | then <math style="vertical-align: -20px">\sum_{n=1}^\infty a_n</math> and <math style="vertical-align: -20px">\sum_{n=1}^\infty b_n</math> either both converge or both diverge. |
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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: - | + | |Let <math style="vertical-align: -19px">a_n=\frac{n^3+7n}{\sqrt{1+n^{10}}}.</math> |
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Revision as of 15:48, 5 March 2017
Test if the following series converges or diverges. Give reasons and clearly state if you are using any standard test.
- 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{n^3+7n}{\sqrt{1+n^{10}}}}
| Foundations: |
|---|
| Limit Comparison Test |
| 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\}} 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. |
| 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 \lim_{n\rightarrow \infty} \frac{a_n}{b_n}=L,} 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 L} is a positive real number, |
| 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} 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 \sum_{n=1}^\infty b_n} either both converge or both diverge. |
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{n^3+7n}{\sqrt{1+n^{10}}}>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{n^3+7n}{\sqrt{1+n^{10}}}.} |
| 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: |
|---|
| Now, 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 \begin{array}{rcl} \displaystyle{\lim_{n\rightarrow \infty} \frac{a_n}{b_n}} & = & \displaystyle{\lim_{n\rightarrow \infty} \frac{(\frac{n^3+7n}{\sqrt{1+n^{10}}})}{(\frac{1}{n^2})}}\\ &&\\ & = & \displaystyle{\lim_{n\rightarrow \infty} \frac{n^5+7n^3}{\sqrt{1+n^{10}}}}\\ &&\\ & = & \displaystyle{\lim_{n\rightarrow \infty} \frac{n^5+7n^3}{\sqrt{1+n^{10}}} \bigg(\frac{\frac{1}{n^5}}{\frac{1}{n^5}}\bigg)}\\ &&\\ & = & \displaystyle{\lim_{n\rightarrow \infty} \frac{1+\frac{7}{n^4}}{\sqrt{\frac{1}{n^{10}}+1}}}\\ &&\\ & = & \displaystyle{1.} \end{array}} |
| 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} \frac{n^3+7n}{\sqrt{1+n^{10}}}} |
| converges by the Limit Comparison Test. |
| Final Answer: |
|---|
| converges |