Nth+Root+Test

__Nth Root Test__

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**What is the Root Test**?  The root test is a test designed to test for convergence of an infinite series. As with the Ratio Test, this test will also tell whether a series is absolutely convergent, rather than just simple convergence. It was developed by the French mathematician Augustin-Louis Cauchy.

**Root Test **
 *  Suppose that we have the series [[image:http://tutorial.math.lamar.edu/Classes/CalcII/RootTest_files/eq0001MP.gif width="46" height="30"]] Define,  [[image:http://tutorial.math.lamar.edu/Classes/CalcII/RootTest_files/eq0002MP.gif width="220" height="47"]] [[image:http://tutorial.math.lamar.edu/Classes/CalcII/RootTest_files/empty.gif width="145" height="26"]][[image:http://tutorial.math.lamar.edu/Classes/CalcII/RootTest_files/empty.gif width="145" height="26"]] [[image:http://tutorial.math.lamar.edu/Classes/CalcII/RootTest_files/empty.gif width="1" height="1" align="top"]]   Then,   1. if [[image:http://tutorial.math.lamar.edu/Classes/CalcII/RootTest_files/eq0003MP.gif width="45" height="15"]] [[image:http://tutorial.math.lamar.edu/Classes/CalcII/RootTest_files/empty.gif width="29" height="15"]][[image:http://tutorial.math.lamar.edu/Classes/CalcII/RootTest_files/empty.gif width="29" height="15"]] [[image:http://tutorial.math.lamar.edu/Classes/CalcII/RootTest_files/empty.gif width="1" height="19" align="top"]][[image:http://tutorial.math.lamar.edu/Classes/CalcII/RootTest_files/empty.gif width="1" height="1" align="bottom"]][[image:http://tutorial.math.lamar.edu/Classes/CalcII/RootTest_files/empty.gif width="1" height="1" align="top"]] the series is absolutely convergent (and hence convergent).   2. if [[image:http://tutorial.math.lamar.edu/Classes/CalcII/RootTest_files/eq0004MP.gif width="45" height="15"]] [[image:http://tutorial.math.lamar.edu/Classes/CalcII/RootTest_files/empty.gif width="29" height="15"]][[image:http://tutorial.math.lamar.edu/Classes/CalcII/RootTest_files/empty.gif width="29" height="15"]] [[image:http://tutorial.math.lamar.edu/Classes/CalcII/RootTest_files/empty.gif width="1" height="19" align="top"]][[image:http://tutorial.math.lamar.edu/Classes/CalcII/RootTest_files/empty.gif width="1" height="1" align="bottom"]][[image:http://tutorial.math.lamar.edu/Classes/CalcII/RootTest_files/empty.gif width="1" height="1" align="top"]] the series is divergent   3. if <span class="MPPopup" style="background-color: transparent; font-family: Times New Roman; font-size: 16px; height: 15px; text-decoration: none; vertical-align: auto; width: 45px;">[[image:http://tutorial.math.lamar.edu/Classes/CalcII/RootTest_files/eq0005MP.gif width="45" height="15"]] <span class="MPPHSpan" style="background-color: transparent; font-family: Times New Roman; font-size: 0px; height: 1px; position: relative; text-decoration: none; vertical-align: auto;">[[image:http://tutorial.math.lamar.edu/Classes/CalcII/RootTest_files/empty.gif width="29" height="15"]][[image:http://tutorial.math.lamar.edu/Classes/CalcII/RootTest_files/empty.gif width="29" height="15"]] [[image:http://tutorial.math.lamar.edu/Classes/CalcII/RootTest_files/empty.gif width="1" height="19" align="top"]][[image:http://tutorial.math.lamar.edu/Classes/CalcII/RootTest_files/empty.gif width="1" height="1" align="bottom"]][[image:http://tutorial.math.lamar.edu/Classes/CalcII/RootTest_files/empty.gif width="1" height="1" align="top"]] the series may be divergent, conditionally convergent, or absolutely convergent.  ||

<span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;"> As with the ratio test, if we get <span class="MPPopup" style="font-family: Times New Roman; height: 15px; text-decoration: none; width: 45px;"> the root test will tell us nothing and we’ll need to use another test to determine the convergence of the series. Also note that if <span class="MPPopup" style="font-family: Times New Roman; height: 15px; text-decoration: none; width: 45px;"> in the Ratio Test then the Root Test will also give <span class="MPPopup" style="font-family: Times New Roman; height: 15px; text-decoration: none; width: 45px;"> <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">**How do you know when to use this test as opposed to others?** <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;"> This can sometimes be the trickiest part of dealing with testing for convergence of series. There are no concrete ways to determine which test you should use, however that doesnt mean you should just start throwing out random tests. There are usually hints you can pick up on from the series that will give you an idea about what test to use. For instance if an infinite series contains a square root of "N", or N raised to some power, such as in the Power Series, then it is almost certain that the Nth root test can be used to determine convergence. <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;"> <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">**__PROOF__** <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;"> <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">**Examples:**
 * If the test is inconclusive, what should you do?**

<span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;"> <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;"> <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">__Practice Problems__ <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">1. <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;"> <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;"> <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;"> <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;"> <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">2. <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;"> <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">3. <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">

<span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">__Solutions to Practice Problems__

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