Series Problems - Revision history

Kayla Murray at 20:43, 22 October 2017

2017-10-22T20:43:03Z

Kayla Murray: /* Problem 6 */

2017-10-22T20:35:02Z

Problem 6

Kayla Murray at 20:34, 22 October 2017

2017-10-22T20:34:46Z

Kayla Murray at 20:30, 22 October 2017

2017-10-22T20:30:02Z

Kayla Murray at 20:24, 22 October 2017

2017-10-22T20:24:44Z

Kayla Murray: Created page with "text"

2017-10-22T20:04:45Z

Created page with "text"

New page

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 ::<span class="exam"><math>\sum_{n=1}^\infty \frac{(n^2+1)^{2n}}{(2n^2+1)^n}</math>
-== [[031_Review Part 1,_Problem_8|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 8&nbsp;</span>]] ==
+== [[Series Problems,_Problem_8|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 8&nbsp;</span>]] ==
-<span class="exam">True or false: Let &nbsp;<math style="vertical-align: 0px">W</math>&nbsp; be a subspace of &nbsp;<math style="vertical-align: 0px">\mathbb{R}^4</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">\vec{v}</math>&nbsp; be a vector in &nbsp;<math style="vertical-align: 0px">\mathbb{R}^4.</math>&nbsp; If &nbsp;<math style="vertical-align: 0px">\vec{v}\in W</math>&nbsp; and &nbsp;<math style="vertical-align: -4px">\vec{v}\in W^\perp,</math>&nbsp; then &nbsp;<math style="vertical-align: 0px">\vec{v}=\vec{0}.</math>
+::<span class="exam"><math>\sum_{n=1}^\infty \frac{6n-12}{n^2-4n+5}</math> (using the Integral Test)
-== [[031_Review Part 1,_Problem_9|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 9&nbsp;</span>]] ==
+== [[Series Problems,_Problem_9|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 9&nbsp;</span>]] ==
-<span class="exam">True or false: If &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is an invertible &nbsp;<math style="vertical-align: 0px">3\times 3</math>&nbsp; matrix, and &nbsp;<math style="vertical-align: 0px">B</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">C</math>&nbsp; are &nbsp;<math style="vertical-align: 0px">3\times 3</math>&nbsp; matrices such that &nbsp;<math style="vertical-align: -4px">AB=AC,</math>&nbsp; then &nbsp;<math style="vertical-align: 0px">B=C.</math>
+::<span class="exam"><math>\sum_{n=1}^\infty \frac{n-1}{n^2\sqrt{n}}</math>

@@ Line 22: / Line 22: @@
 == [[Series Problems,_Problem_6|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 6&nbsp;</span>]] ==
-::<span class="exam"><math>\sum_{n=1}^\infty \frac{10^n}{(n+1)4^{2n+1}}
+::<span class="exam"><math>\sum_{n=1}^\infty \frac{10^n}{(n+1)4^{2n+1}}</math>
 == [[Series Problems,_Problem_7|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 7&nbsp;</span>]] ==

@@ Line 15: / Line 15: @@
 ::<span class="exam"><math>\sum_{n=1}^\infty \frac{n}{n^4+1}</math>
-== [[031_Review Part 1,_Problem_4|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 4&nbsp;</span>]] ==
+== [[Series Problems,_Problem_4|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 4&nbsp;</span>]] ==
-<span class="exam"> True or false: If &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is invertible, then &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is diagonalizable.
+::<span class="exam"><math>\sum_{n=1}^\infty \frac{n^2-1}{3n^4+1}</math>
-== [[031_Review Part 1,_Problem_5|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 5&nbsp;</span>]] ==
+== [[Series Problems,_Problem_5|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 5&nbsp;</span>]] ==
-<span class="exam">True or false: If &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">B</math>&nbsp; are invertible &nbsp;<math style="vertical-align: 0px">n\times n</math>&nbsp; matrices, then so is &nbsp;<math style="vertical-align: -1px">A+B.</math>
+::<span class="exam"><math>\sum_{n=1}^\infty \frac{(-1)^{n-1}n^2}{10^n}</math>
-== [[031_Review Part 1,_Problem_6|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 6&nbsp;</span>]] ==
+== [[Series Problems,_Problem_6|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 6&nbsp;</span>]] ==
-<span class="exam"> True or false: If &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is a &nbsp;<math style="vertical-align: 0px">3\times 5</math>&nbsp; matrix and &nbsp;<math style="vertical-align: -4px">\text{dim Nul }A=2,</math>&nbsp; then &nbsp;<math style="vertical-align: 0px">A\vec{x}=\vec{b}</math>&nbsp; is consistent for all &nbsp;<math style="vertical-align: 0px">\vec{b}</math>&nbsp; in &nbsp;<math style="vertical-align: 0px">\mathbb{R}^3.</math>
+::<span class="exam"><math>\sum_{n=1}^\infty \frac{10^n}{(n+1)4^{2n+1}}
-== [[031_Review Part 1,_Problem_7|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 7&nbsp;</span>]] ==
+== [[Series Problems,_Problem_7|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 7&nbsp;</span>]] ==
+::<span class="exam"><math>\sum_{n=1}^\infty \frac{(n^2+1)^{2n}}{(2n^2+1)^n}</math>
 == [[031_Review Part 1,_Problem_8|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 8&nbsp;</span>]] ==

@@ Line 1: / Line 1: @@
-'''These questions are meant to be additional practice problems for series.'''
+'''These questions are meant to be practice problems for series.'''
 '''Determine whether the series converge or diverge.'''
@@ Line 6: / Line 6: @@
 <div class="noautonum">__TOC__</div>
-== [[031_Review Part 1,_Problem_1|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 1&nbsp;</span></span>]] ==
+== [[Series Problems,_Problem_1|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 1&nbsp;</span></span>]] ==
-<span class="exam">True or false: If all the entries of a &nbsp;<math style="vertical-align: 0px">7\times 7</math>&nbsp; matrix &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; are &nbsp;<math style="vertical-align: -4px">7,</math>&nbsp; then &nbsp;<math style="vertical-align: 0px">\text{det }A</math>&nbsp; must be &nbsp;<math style="vertical-align: 0px">7^7.</math>
+::<span class="exam"><math>\sum_{n=1}^\infty \frac{2+3^n}{4^n}</math>
-== [[031_Review Part 1,_Problem_2|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 2&nbsp;</span>]] ==
+== [[Series Problems,_Problem_2|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 2&nbsp;</span>]] ==
-<span class="exam"> True or false: If a matrix &nbsp;<math style="vertical-align: 0px">A^2</math>&nbsp; is diagonalizable, then the matrix &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; must be diagonalizable as well.
+::<span class="exam"><math>\sum_{n=1}^\infty \ln\Bigg(\frac{n^2+1}{2n^2+1}\Bigg)</math>
-== [[031_Review Part 1,_Problem_3|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 3&nbsp;</span>]] ==
+== [[Series Problems,_Problem_3|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 3&nbsp;</span>]] ==
-<span class="exam">True or false: If &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is a &nbsp;<math style="vertical-align: -1px">4\times 4</math>&nbsp; matrix with characteristic equation &nbsp;<math style="vertical-align: -5px">\lambda(\lambda-1)(\lambda+1)(\lambda+e)=0,</math>&nbsp; then &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is diagonalizable.
+::<span class="exam"><math>\sum_{n=1}^\infty \frac{n}{n^4+1}</math>
 == [[031_Review Part 1,_Problem_4|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 4&nbsp;</span>]] ==

← Older revision		Revision as of 20:24, 22 October 2017
Line 1:		Line 1:
−	text	+	'''These questions are meant to be additional practice problems for series.'''
		+
		+	'''Determine whether the series converge or diverge.'''
		+
		+	'''Click on the <span class="biglink" style="color:darkblue;"> boxed problem numbers </span> to go to a solution.'''
		+	<div class="noautonum">__TOC__</div>
		+
		+	== [[031_Review Part 1,_Problem_1\|<span class="biglink"><span style="font-size:80%"> Problem 1 </span></span>]] ==
		+	<span class="exam">True or false: If all the entries of a  <math style="vertical-align: 0px">7\times 7</math>  matrix  <math style="vertical-align: 0px">A</math>  are  <math style="vertical-align: -4px">7,</math>  then  <math style="vertical-align: 0px">\text{det }A</math>  must be  <math style="vertical-align: 0px">7^7.</math>
		+
		+	== [[031_Review Part 1,_Problem_2\|<span class="biglink"><span style="font-size:80%"> Problem 2 </span>]] ==
		+	<span class="exam"> True or false: If a matrix  <math style="vertical-align: 0px">A^2</math>  is diagonalizable, then the matrix  <math style="vertical-align: 0px">A</math>  must be diagonalizable as well.
		+
		+	== [[031_Review Part 1,_Problem_3\|<span class="biglink"><span style="font-size:80%"> Problem 3 </span>]] ==
		+	<span class="exam">True or false: If  <math style="vertical-align: 0px">A</math>  is a  <math style="vertical-align: -1px">4\times 4</math>  matrix with characteristic equation  <math style="vertical-align: -5px">\lambda(\lambda-1)(\lambda+1)(\lambda+e)=0,</math>  then  <math style="vertical-align: 0px">A</math>  is diagonalizable.
		+
		+	== [[031_Review Part 1,_Problem_4\|<span class="biglink"><span style="font-size:80%"> Problem 4 </span>]] ==
		+	<span class="exam"> True or false: If  <math style="vertical-align: 0px">A</math>  is invertible, then  <math style="vertical-align: 0px">A</math>  is diagonalizable.
		+
		+	== [[031_Review Part 1,_Problem_5\|<span class="biglink"><span style="font-size:80%"> Problem 5 </span>]] ==
		+	<span class="exam">True or false: If  <math style="vertical-align: 0px">A</math>  and  <math style="vertical-align: 0px">B</math>  are invertible  <math style="vertical-align: 0px">n\times n</math>  matrices, then so is  <math style="vertical-align: -1px">A+B.</math>
		+
		+	== [[031_Review Part 1,_Problem_6\|<span class="biglink"><span style="font-size:80%"> Problem 6 </span>]] ==
		+	<span class="exam"> True or false: If  <math style="vertical-align: 0px">A</math>  is a  <math style="vertical-align: 0px">3\times 5</math>  matrix and  <math style="vertical-align: -4px">\text{dim Nul }A=2,</math>  then  <math style="vertical-align: 0px">A\vec{x}=\vec{b}</math>  is consistent for all  <math style="vertical-align: 0px">\vec{b}</math>  in  <math style="vertical-align: 0px">\mathbb{R}^3.</math>
		+
		+	== [[031_Review Part 1,_Problem_7\|<span class="biglink"><span style="font-size:80%"> Problem 7 </span>]] ==
		+
		+	<span class="exam">True or false: Let  <math style="vertical-align: 0px">C=AB</math>  for  <math style="vertical-align: 0px">4\times 4</math>  matrices  <math style="vertical-align: 0px">A</math>  and  <math style="vertical-align: 0px">B.</math>  If  <math style="vertical-align: 0px">C</math>  is invertible, then  <math style="vertical-align: 0px">A</math>  is invertible.
		+
		+	== [[031_Review Part 1,_Problem_8\|<span class="biglink"><span style="font-size:80%"> Problem 8 </span>]] ==
		+
		+	<span class="exam">True or false: Let  <math style="vertical-align: 0px">W</math>  be a subspace of  <math style="vertical-align: 0px">\mathbb{R}^4</math>  and  <math style="vertical-align: 0px">\vec{v}</math>  be a vector in  <math style="vertical-align: 0px">\mathbb{R}^4.</math>  If  <math style="vertical-align: 0px">\vec{v}\in W</math>  and  <math style="vertical-align: -4px">\vec{v}\in W^\perp,</math>  then  <math style="vertical-align: 0px">\vec{v}=\vec{0}.</math>
		+
		+	== [[031_Review Part 1,_Problem_9\|<span class="biglink"><span style="font-size:80%"> Problem 9 </span>]] ==
		+
		+	<span class="exam">True or false: If  <math style="vertical-align: 0px">A</math>  is an invertible  <math style="vertical-align: 0px">3\times 3</math>  matrix, and  <math style="vertical-align: 0px">B</math>  and  <math style="vertical-align: 0px">C</math>  are  <math style="vertical-align: 0px">3\times 3</math>  matrices such that  <math style="vertical-align: -4px">AB=AC,</math>  then  <math style="vertical-align: 0px">B=C.</math>