Difference between revisions of "009C Sample Final 2, Problem 3"
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| − | <span class="exam">Determine if the following series converges or diverges. Please give your reason(s). | + | <span class="exam"> Determine if the following series converges or diverges. Please give your reason(s). |
| − | <span class="exam">(a) <math>\sum_{n=1}^{\infty} \frac{n!}{(2n)!}</math> | + | <span class="exam">(a) <math>\sum_{n=1}^{\infty} \frac{n!}{(2n)!}</math> |
| − | <span class="exam">(b) <math>\sum_{n=1}^{\infty} (-1)^n \frac{1}{n+1}</math> | + | <span class="exam">(b) <math>\sum_{n=1}^{\infty} (-1)^n\frac{1}{n+1}</math> |
{| class="mw-collapsible mw-collapsed" style = "text-align:left;" | {| class="mw-collapsible mw-collapsed" style = "text-align:left;" | ||
| Line 48: | Line 48: | ||
| | | | ||
<math>\begin{array}{rcl} | <math>\begin{array}{rcl} | ||
| − | \displaystyle{\lim_{n\rightarrow \infty}\bigg|\frac{a_{n+1}}{a_n}\bigg|} & = & \displaystyle{\lim_{n\rightarrow \infty} \bigg| \frac{(n+1)!}{(2(n+1))!} | + | \displaystyle{\lim_{n\rightarrow \infty}\bigg|\frac{a_{n+1}}{a_n}\bigg|} & = & \displaystyle{\lim_{n\rightarrow \infty} \bigg| \frac{(n+1)!}{(2(n+1))!} \frac{(2n)!}{n!}\bigg|}\\ |
&&\\ | &&\\ | ||
| − | & = & \displaystyle{\lim_{n\rightarrow \infty} \bigg| \frac{(n+1)n!}{(2n+2)(2n+1)(2n)!} | + | & = & \displaystyle{\lim_{n\rightarrow \infty} \bigg| \frac{(n+1)n!}{(2n+2)(2n+1)(2n)!} \frac{(2n)!}{n!}\bigg|}\\ |
&&\\ | &&\\ | ||
& = & \displaystyle{\lim_{n\rightarrow \infty} \frac{n+1}{(2n+2)(2n+1)}}\\ | & = & \displaystyle{\lim_{n\rightarrow \infty} \frac{n+1}{(2n+2)(2n+1)}}\\ | ||
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|- | |- | ||
|Let <math style="vertical-align: -16px"> b_n=\frac{1}{n+1}.</math> | |Let <math style="vertical-align: -16px"> b_n=\frac{1}{n+1}.</math> | ||
| + | |- | ||
| + | |First, we have | ||
| + | |- | ||
| + | | <math>\frac{1}{n+1}\ge 0</math> | ||
| + | |- | ||
| + | |for all <math style="vertical-align: -3px">n\ge 1.</math> | ||
|- | |- | ||
|The sequence <math style="vertical-align: -5px">\{b_n\}</math> is decreasing since | |The sequence <math style="vertical-align: -5px">\{b_n\}</math> is decreasing since | ||
| Line 87: | Line 93: | ||
| <math>\frac{1}{n+2}<\frac{1}{n+1}</math> | | <math>\frac{1}{n+2}<\frac{1}{n+1}</math> | ||
|- | |- | ||
| − | |for all <math style="vertical-align: -3px">n\ge | + | |for all <math style="vertical-align: -3px">n\ge 1.</math> |
|} | |} | ||
| Line 106: | Line 112: | ||
!Final Answer: | !Final Answer: | ||
|- | |- | ||
| − | | '''(a)''' converges | + | | '''(a)''' converges (by the Ratio Test) |
|- | |- | ||
| − | | '''(b)''' converges | + | | '''(b)''' converges (by the Alternating Series Test) |
|} | |} | ||
[[009C_Sample_Final_2|'''<u>Return to Sample Exam</u>''']] | [[009C_Sample_Final_2|'''<u>Return to Sample Exam</u>''']] | ||
Revision as of 11:16, 17 March 2017
Determine if the following series converges or diverges. Please give your reason(s).
(a)
(b) 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} (-1)^n\frac{1}{n+1}}
| Foundations: |
|---|
| 1. Ratio 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 \sum a_n} be a series 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 L=\lim_{n\rightarrow \infty}\bigg|\frac{a_{n+1}}{a_n}\bigg|.} |
| Then, |
|
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 L<1,} the series is absolutely convergent. |
|
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 L>1,} the series is divergent. |
|
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 L=1,} the test is inconclusive. |
| 2. If a series absolutely converges, then it also converges. |
| 3. Alternating Series 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\}} be a positive, decreasing sequence 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 \lim_{n\rightarrow \infty} a_n=0.} |
| 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 (-1)^na_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 (-1)^{n+1}a_n} |
| converge. |
Solution:
(a)
| Step 1: |
|---|
| We begin by using the Ratio Test. |
| 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}\bigg|\frac{a_{n+1}}{a_n}\bigg|} & = & \displaystyle{\lim_{n\rightarrow \infty} \bigg| \frac{(n+1)!}{(2(n+1))!} \frac{(2n)!}{n!}\bigg|}\\ &&\\ & = & \displaystyle{\lim_{n\rightarrow \infty} \bigg| \frac{(n+1)n!}{(2n+2)(2n+1)(2n)!} \frac{(2n)!}{n!}\bigg|}\\ &&\\ & = & \displaystyle{\lim_{n\rightarrow \infty} \frac{n+1}{(2n+2)(2n+1)}}\\ &&\\ & = & \displaystyle{0.} \end{array}} |
| Step 2: |
|---|
| 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 \lim_{n\rightarrow \infty} \bigg|\frac{a_{n+1}}{a_n}\bigg|=0<1,} |
| the series is absolutely convergent by the Ratio Test. |
| Therefore, the series converges. |
(b)
| Step 1: |
|---|
| For |
| 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 (-1)^n\frac{1}{n+1},} |
| we notice that this series is alternating. |
| 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+1}.} |
| First, 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 \frac{1}{n+1}\ge 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.} |
| The sequence 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\}} is decreasing 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+2}<\frac{1}{n+1}} |
| 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.} |
| Step 2: |
|---|
| Also, |
| 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}b_n=\lim_{n\rightarrow \infty}\frac{1}{n+1}=0.} |
| 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 (-1)^n\frac{1}{n+1}} converges |
| by the Alternating Series Test. |
| Final Answer: |
|---|
| (a) converges (by the Ratio Test) |
| (b) converges (by the Alternating Series Test) |