Difference between revisions of "009C Sample Final 2, Problem 3"
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!Step 1: | !Step 1: | ||
|- | |- | ||
| − | | | + | |For |
|- | |- | ||
| − | | | + | | <math>\sum_{n=1}^\infty (-1)^n\frac{1}{n+1},</math> |
|- | |- | ||
| − | | | + | |we notice that this series is alternating. |
| + | |- | ||
| + | |Let <math style="vertical-align: -16px"> b_n=\frac{1}{n+1}.</math> | ||
| + | |- | ||
| + | |The sequence <math style="vertical-align: -5px">\{b_n\}</math> is decreasing since | ||
| + | |- | ||
| + | | <math>\frac{1}{n+2}<\frac{1}{n+1}</math> | ||
| + | |- | ||
| + | |for all <math style="vertical-align: -3px">n\ge 0.</math> | ||
|} | |} | ||
| Line 57: | Line 65: | ||
!Step 2: | !Step 2: | ||
|- | |- | ||
| − | | | + | |Also, |
| + | |- | ||
| + | | <math>\lim_{n\rightarrow \infty}b_n=\lim_{n\rightarrow \infty}\frac{1}{n+1}=0.</math> | ||
| + | |- | ||
| + | |Therefore, the series <math>\sum_{n=1}^\infty (-1)^n\frac{1}{n+1}</math> converges | ||
|- | |- | ||
| − | | | + | |by the Alternating Series Test. |
|} | |} | ||
| Line 68: | Line 80: | ||
| '''(a)''' | | '''(a)''' | ||
|- | |- | ||
| − | | '''(b)''' | + | | '''(b)''' converges |
|} | |} | ||
[[009C_Sample_Final_2|'''<u>Return to Sample Exam</u>''']] | [[009C_Sample_Final_2|'''<u>Return to Sample Exam</u>''']] | ||
Revision as of 20:40, 4 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=0}^{+\infty} (-1)^n \frac{1}{n+1}}
| Foundations: |
|---|
Solution:
(a)
| Step 1: |
|---|
| Step 2: |
|---|
(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}.} |
| 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 0.} |
| 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) |
| (b) converges |