Difference between revisions of "009C Sample Midterm 2, Problem 5"
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!Foundations: | !Foundations: | ||
|- | |- | ||
| − | |A geometric series <math>\sum_{n=0}^{\infty} ar^n</math> converges if <math>|r|<1.</math> | + | |A geometric series <math>\sum_{n=0}^{\infty} ar^n</math> converges if <math style="vertical-align: -6px">|r|<1.</math> |
|} | |} | ||
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|First, we notice that <math>\sum_{n=0}^\infty c_nx^n</math> is a geometric series. | |First, we notice that <math>\sum_{n=0}^\infty c_nx^n</math> is a geometric series. | ||
|- | |- | ||
| − | |We have <math>r=x.</math> | + | |We have <math style="vertical-align: -1px">r=x.</math> |
|- | |- | ||
|Since this series converges, | |Since this series converges, | ||
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|The series <math>\sum_{n=0} c_n\bigg(\frac{x}{2}\bigg)^n</math> is also a geometric series. | |The series <math>\sum_{n=0} c_n\bigg(\frac{x}{2}\bigg)^n</math> is also a geometric series. | ||
|- | |- | ||
| − | |For this series, <math>r=\frac{x}{2}.</math> | + | |For this series, <math style="vertical-align: -13px">r=\frac{x}{2}.</math> |
|- | |- | ||
|Now, we notice | |Now, we notice | ||
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\end{array}</math> | \end{array}</math> | ||
|- | |- | ||
| − | |since <math>|x|<1.</math> | + | |since <math style="vertical-align: -5px">|x|<1.</math> |
|- | |- | ||
| − | | Since <math>|r|<1,</math> this series converges. | + | | Since <math style="vertical-align: -5px">|r|<1,</math> this series converges. |
|} | |} | ||
| Line 57: | Line 57: | ||
|First, we notice that <math>\sum_{n=0}^\infty c_nx^n</math> is a geometric series. | |First, we notice that <math>\sum_{n=0}^\infty c_nx^n</math> is a geometric series. | ||
|- | |- | ||
| − | |We have <math>r=x.</math> | + | |We have <math style="vertical-align: -1px">r=x.</math> |
|- | |- | ||
|Since this series converges, | |Since this series converges, | ||
| Line 69: | Line 69: | ||
|The series <math>\sum_{n=0}^\infty c_n(-x)^n</math> is also a geometric series. | |The series <math>\sum_{n=0}^\infty c_n(-x)^n</math> is also a geometric series. | ||
|- | |- | ||
| − | |For this series, <math>r=-x.</math> | + | |For this series, <math style="vertical-align: -1px">r=-x.</math> |
|- | |- | ||
|Now, we notice | |Now, we notice | ||
| Line 82: | Line 82: | ||
\end{array}</math> | \end{array}</math> | ||
|- | |- | ||
| − | |since <math>|x|<1.</math> | + | |since <math style="vertical-align: -5px">|x|<1.</math> |
|- | |- | ||
| − | |Since <math>|r|<1,</math> this series converges. | + | |Since <math style="vertical-align: -5px">|r|<1,</math> this series converges. |
|} | |} | ||
Revision as of 16:51, 15 February 2017
If converges, does it follow that the following series converges?
- a)
- b)
| Foundations: |
|---|
| A geometric 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=0}^{\infty} ar^n} converges 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 |r|<1.} |
Solution:
(a)
| Step 1: |
|---|
| First, we notice 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 \sum_{n=0}^\infty c_nx^n} is a geometric series. |
| 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 r=x.} |
| Since this series converges, |
| 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 |r|=|x|<1.} |
| Step 2: |
|---|
| 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=0} c_n\bigg(\frac{x}{2}\bigg)^n} is also a geometric series. |
| For this 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 r=\frac{x}{2}.} |
| Now, we notice |
|
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{|r|} & = & \displaystyle{\bigg|\frac{x}{2}\bigg|}\\ &&\\ & = & \displaystyle{\frac{|x|}{2}}\\ &&\\ & < & \displaystyle{\frac{1}{2}} \end{array}} |
| 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 |x|<1.} |
| 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 |r|<1,} this series converges. |
(b)
| Step 1: |
|---|
| First, we notice 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 \sum_{n=0}^\infty c_nx^n} is a geometric series. |
| 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 r=x.} |
| Since this series converges, |
| 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 |r|=|x|<1.} |
| Step 2: |
|---|
| 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=0}^\infty c_n(-x)^n} is also a geometric series. |
| For this 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 r=-x.} |
| Now, we notice |
|
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{|r|} & = & \displaystyle{|-x|}\\ &&\\ & = & \displaystyle{|x|}\\ &&\\ & < & \displaystyle{1} \end{array}} |
| 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 |x|<1.} |
| 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 |r|<1,} this series converges. |
| Final Answer: |
|---|
| (a) The series converges. |
| (b) The series converges. |