https://mathresearch.utsa.edu/wiki/index.php?action=history&feed=atom&title=Ratio_and_Root_Tests 2026-06-12T21:03:36Z Revision history for this page on the wiki MediaWiki 1.34.1 https://mathresearch.utsa.edu/wiki/index.php?title=Ratio_and_Root_Tests&diff=3268&oldid=prev 2021-10-29T19:34:07Z

<p><span dir="auto"><span class="autocomment">Licensing</span></span></p> <table class="diff diff-contentalign-left" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 19:34, 29 October 2021</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l209" >Line 209:</td> <td colspan="2" class="diff-lineno">Line 209:</td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Content obtained and/or adapted from:</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Content obtained and/or adapted from:</div></td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* [https://en.wikipedia.org/wiki/Root_test Root test, Wikipedia] under a CC BY-SA license</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* [https://en.wikipedia.org/wiki/Root_test Root test, Wikipedia] under a CC BY-SA license</div></td></tr> <tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* [https://en.wikipedia.org/wiki/Ratio_test Ratio test, Wikipedia] under a CC BY-SA license</ins></div></td></tr> </table>

Lila https://mathresearch.utsa.edu/wiki/index.php?title=Ratio_and_Root_Tests&diff=3267&oldid=prev 2021-10-29T19:32:47Z

<p><span dir="auto"><span class="autocomment">References</span></span></p> <table class="diff diff-contentalign-left" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 19:32, 29 October 2021</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l205" >Line 205:</td> <td colspan="2" class="diff-lineno">Line 205:</td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>#Knopp, Konrad (1956). &quot;§ 3.2&quot;. Infinite Sequences and Series. Dover publications, Inc., New York. ISBN 0-486-60153-6.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>#Knopp, Konrad (1956). &quot;§ 3.2&quot;. Infinite Sequences and Series. Dover publications, Inc., New York. ISBN 0-486-60153-6.</div></td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>#Whittaker, E. T. &amp; Watson, G. N. (1963). &quot;§ 2.35&quot;. A Course in Modern Analysis (fourth ed.). Cambridge University Press. ISBN 0-521-58807-3.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>#Whittaker, E. T. &amp; Watson, G. N. (1963). &quot;§ 2.35&quot;. A Course in Modern Analysis (fourth ed.). Cambridge University Press. ISBN 0-521-58807-3.</div></td></tr> <tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==Licensing==</ins></div></td></tr> <tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Content obtained and/or adapted from:</ins></div></td></tr> <tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* [https://en.wikipedia.org/wiki/Root_test Root test, Wikipedia] under a CC BY-SA license</ins></div></td></tr> </table>

Lila https://mathresearch.utsa.edu/wiki/index.php?title=Ratio_and_Root_Tests&diff=2064&oldid=prev 2021-10-10T21:17:07Z

<p></p> <table class="diff diff-contentalign-left" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 21:17, 10 October 2021</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l144" >Line 144:</td> <td colspan="2" class="diff-lineno">Line 144:</td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Applying the root test and using the fact that &lt;math&gt; \lim_{n \rightarrow \infty} n^{1/n}=1,&lt;/math&gt;</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Applying the root test and using the fact that &lt;math&gt; \lim_{n \rightarrow \infty} n^{1/n}=1,&lt;/math&gt;</div></td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>::&lt;math&gt; C = \sqrt[n]{|\frac{2^n}{n^9}|}= \frac{ \sqrt[n]{2^n} } { \sqrt[n]{n^9} } = \frac{ 2 }  {(n^{1/n})^9 }  = 2 &lt;/math&gt;  </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>::&lt;math&gt; C = \sqrt[n]{|\frac{2^n}{n^9}|}= \frac{ \sqrt[n]{2^n} } { \sqrt[n]{n^9} } = \frac{ 2 }  {(n^{1/n})^9 }  = 2 &lt;/math&gt;  </div></td></tr> <tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Since &lt;math&gt; C=2&gt;1,&lt;/math&gt;  the series diverges.<del class="diffchange diffchange-inline">&lt;ref&gt;{{cite book</del></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Since &lt;math&gt; C=2&gt;1,&lt;/math&gt;  the series diverges.</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline"> | first1= William |last1= Briggs|first2= Lyle|last2 = Cochrane</del></div></td><td colspan="2"> </td></tr> <tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline"> | title= Calculus: Early Transcendentals</del></div></td><td colspan="2"> </td></tr> <tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline"> | url= https://archive.org/details/calculusearlytra0000brig/page/n5/mode/2up</del></div></td><td colspan="2"> </td></tr> <tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline"> | url-access= registration</del></div></td><td colspan="2"> </td></tr> <tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline"> | publisher= Addison Wesley</del></div></td><td colspan="2"> </td></tr> <tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline"> | year=2011</del></div></td><td colspan="2"> </td></tr> <tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline"> }}  p. 571. &lt;/ref&gt;</del></div></td><td colspan="2"> </td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>''Example 2:''  </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>''Example 2:''  </div></td></tr> </table>

Khanh https://mathresearch.utsa.edu/wiki/index.php?title=Ratio_and_Root_Tests&diff=2063&oldid=prev 2021-10-10T21:16:11Z

<p></p> <table class="diff diff-contentalign-left" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 21:16, 10 October 2021</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l157" >Line 157:</td> <td colspan="2" class="diff-lineno">Line 157:</td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The root test shows convergence because   </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The root test shows convergence because   </div></td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:: &lt;math&gt;r=\limsup_{n\to\infty}\sqrt[n]{|a_n|} = \limsup_{n\to\infty}\sqrt[n]{|.5^n|}=.5&lt;1.&lt;/math&gt;</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:: &lt;math&gt;r=\limsup_{n\to\infty}\sqrt[n]{|a_n|} = \limsup_{n\to\infty}\sqrt[n]{|.5^n|}=.5&lt;1.&lt;/math&gt;</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>This example shows how the root test is stronger than the <del class="diffchange diffchange-inline">[[</del>ratio test<del class="diffchange diffchange-inline">]]</del>. The ratio test  is inconclusive for this series if &lt;math&gt;n&lt;/math&gt; is odd so &lt;math&gt;a_n=a_{n+1} = .5^n&lt;/math&gt; (though not if &lt;math&gt;n&lt;/math&gt; is even), because   </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>This example shows how the root test is stronger than the ratio test. The ratio test  is inconclusive for this series if &lt;math&gt;n&lt;/math&gt; is odd so &lt;math&gt;a_n=a_{n+1} = .5^n&lt;/math&gt; (though not if &lt;math&gt;n&lt;/math&gt; is even), because   </div></td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:: &lt;math&gt;r=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right| = \lim_{n\to\infty}\left|\frac{ 2  \cdot.5^{n}}{2 \cdot.5^{n}}\right| =1. &lt;/math&gt;</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:: &lt;math&gt;r=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right| = \lim_{n\to\infty}\left|\frac{ 2  \cdot.5^{n}}{2 \cdot.5^{n}}\right| =1. &lt;/math&gt;</div></td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr> </table>

Khanh https://mathresearch.utsa.edu/wiki/index.php?title=Ratio_and_Root_Tests&diff=2062&oldid=prev 2021-10-10T21:14:44Z

<p></p> <a href="https://mathresearch.utsa.edu/wiki/index.php?title=Ratio_and_Root_Tests&amp;diff=2062&amp;oldid=822">Show changes</a>

Khanh https://mathresearch.utsa.edu/wiki/index.php?title=Ratio_and_Root_Tests&diff=822&oldid=prev 2020-11-12T13:32:37Z

<p>Created page with &quot;&lt;strong&gt;The Ratio Test&lt;/strong&gt; * [https://www.youtube.com/watch?v=76BzmM9zDIA The Ratio Test] Video by James Sousa, Math is Power 4U * [https://www.youtube.com/watch?v=SH0dN...&quot;</p> <p><b>New page</b></p><div>&lt;strong&gt;The Ratio Test&lt;/strong&gt;<br /> <br /> * [https://www.youtube.com/watch?v=76BzmM9zDIA The Ratio Test] Video by James Sousa, Math is Power 4U<br /> * [https://www.youtube.com/watch?v=SH0dNW1YMws The Ratio Test I] Video by James Sousa, Math is Power 4U<br /> * [https://www.youtube.com/watch?v=dcGddKmwD_U The Ratio Test II] Video by James Sousa, Math is Power 4U<br /> * [https://www.youtube.com/watch?v=L6p-djs576A Ex 1: The Ratio Test] Video by James Sousa, Math is Power 4U<br /> * [https://www.youtube.com/watch?v=S1BJ16jDiTI Ex 2: The Ratio Test] Video by James Sousa, Math is Power 4U<br /> * [https://www.youtube.com/watch?v=WJdR6PIYE7E Ex 3: The Ratio Test] Video by James Sousa, Math is Power 4U<br /> * [https://www.youtube.com/watch?v=17zAz2KjGfk Ex 4: The Ratio Test] Video by James Sousa, Math is Power 4U<br /> <br /> * [https://www.youtube.com/watch?v=iy8mhbZTY7g Using the Ratio Test to Determine if a Series Converges #1] Video by Patrick JMT<br /> * [https://www.youtube.com/watch?v=gay5CEXnkhA Using the Ratio Test to Determine if a Series Converges #2] Video by Patrick JMT<br /> * [https://www.youtube.com/watch?v=AwJ0P8B25tY Using the Ratio Test to Determine if a Series Converges #3] Video by Patrick JMT<br /> * [https://www.youtube.com/watch?v=nFsoA_dArn4 The Ratio Test - Another Example #1] Video by Patrick JMT<br /> * [https://www.youtube.com/watch?v=KjAkB_2XHaQ The Ratio Test - Another Example #2] Video by Patrick JMT<br /> * [https://www.youtube.com/watch?v=4H57ndz-FRA The Ratio Test - Another Example #3] Video by Patrick JMT<br /> * [https://www.youtube.com/watch?v=U5a3Ls4ZhyA The Ratio Test - Another Example #4] Video by Patrick JMT<br /> <br /> * [https://www.youtube.com/watch?v=WeFse6u19fA The Ratio Test] Video by Krista King<br /> * [https://www.youtube.com/watch?v=uMLBA6CHR1E The Ratio Test with Factorials] Video by Krista King<br /> <br /> * [https://www.youtube.com/watch?v=kkfILYpkJS8 The Ratio Test] Video by The Organic Chemistry Tutor<br /> <br /> <br /> &lt;strong&gt;The Root Test&lt;/strong&gt;<br /> <br /> * [https://www.youtube.com/watch?v=ZE5Pj7s9PWg The Root Test] Video by Math is Power 4U<br /> * [https://www.youtube.com/watch?v=eHzBsFxBzpA The Root Test I] Video by Math is Power 4U<br /> * [https://www.youtube.com/watch?v=eMMtyG7hIyI The Root Test II] Video by Math is Power 4U<br /> * [https://www.youtube.com/watch?v=yVva1LwaE9c Ex 1 The Root Test] Video by Math is Power 4U<br /> * [https://www.youtube.com/watch?v=oKUfBJF5XTI Ex 2 The Root Test] Video by Math is Power 4U<br /> * [https://www.youtube.com/watch?v=Xxzs4sa_zZg Ex 3 The Root Test] Video by Math is Power 4U<br /> * [https://www.youtube.com/watch?v=5HLqCSEgaFI Ex 4 The Root Test] Video by Math is Power 4U<br /> * [https://www.youtube.com/watch?v=JEDrZfwsW1M Ex 5 The Root Test] Video by Math is Power 4U<br /> <br /> * [https://www.youtube.com/watch?v=vDdDLfIya0I The Root Test for Series] Video by Patrick JMT<br /> * [https://www.youtube.com/watch?v=nTfJCbqHyxI The Root Test Another Example 1] Video by Patrick JMT<br /> * [https://www.youtube.com/watch?v=s2ox6IuKiX0 The Root Test Another Example 2] Video by Patrick JMT<br /> * [https://www.youtube.com/watch?v=wo4c4NYIiIo The Root Test Another Example 3] Video by Patrick JMT<br /> <br /> * [https://www.youtube.com/watch?v=pSvQaiqTajk The Root Test] Video by Krista King<br /> <br /> * [https://www.youtube.com/watch?v=ahf0eXOll1M The Root Test] Video by The Organic Chemistry Tutor</div>

Glenn.lahodny