Algebra

Question

using the index of a series as the domain and the value of the seriesas the range is a series a function?

Answer

<P>Hello Debra,<BR><BR>I would say yes.&nbsp;&nbsp;A series is a function over the natural numbers&nbsp;(if the first term is a_0 which is the usual convention).&nbsp;&nbsp;&nbsp;&nbsp;<BR><BR>So let's say f(0) = 4 and f(n) = f(n-1) + (-1)^n * 4/(2n+1) for&nbsp;n &gt; 0.<BR>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;f(1) = 4 - 4/3 ~= 2.667<BR>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; f(2) ~= 2.667 + 4/5 ~= 3.467<BR>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; f(3) ~= 3.467 - 4/7 ~= 2.895<BR><BR>If you have a spreadsheet program you can figure out that&nbsp;<BR><BR>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; f(10000)&nbsp;~= 3.141693<BR><BR>You might guess that<BR>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<BR>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; f(infinity) = pi<BR><BR>And you'd be right.<BR><BR>This series is&nbsp;a variation on the &nbsp;"Madhava-Leibniz-Gregory" series, discovered by Madhava around A.D. 1400 and &nbsp;independently by Gottfried Wilhelm Leibniz and James Gregory around A.D. 1670.&nbsp; <BR><BR>My&nbsp;conventions:&nbsp; _ indicates&nbsp;subscript, ^ indicates&nbsp;exponentiation, and ~= is "approximately equal."<BR><BR>Joe McC.<BR>Forest Inventory and Analysis<BR>Knoxville, TN<BR><BR><BR>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <BR><BR><BR></P> http://vrd.askvrd.org/services/answerschema.xml