seqexs - Examples of Sequences Example 1: sn = In[1]:= n2 +...

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Examples of Sequences Example 1: s n = n 2 + n - n + Sin H 5 n L ê n In[1]:= s @ n_ D = Sqrt @ n^2 + n D - n + Sin @ 5 * n D ê n; Table @ s @ n D êê N, 8 n, 1, 40 <D 8 - 0.544711, 0.177479, 0.680864, 0.700372, 0.450755, 0.316069, 0.422146, 0.578421, 0.581378, 0.461851, 0.398238, 0.464595, 0.55434, 0.546655, 0.466081, 0.430304, 0.482498, 0.542908, 0.52955, 0.468583, 0.447969, 0.492433, 0.535786, 0.51909, 0.470456, 0.459509, 0.498727, 0.530622, 0.511892, 0.472072, 0.46786, 0.503011, 0.526505, 0.506572, 0.473589, 0.474321, 0.506035, 0.523011, 0.502462, 0.475081 < In[2]:= ListPlot @ Table @8 n, s @ n D< , 8 n, 1, 40 <D , AxesOrigin Ø 8 0, 0.2` < , AxesLabel Ø 8 "n", "s n " < , Ticks Ø 8 None, Automatic <D Out[2]= n 0.3 0.4 0.5 0.6 0.7 s n It appears that s n converges to 1 2 . In[3]:= ListPlot @ Table @8 n, s @ n D< , 8 n, 1, 400 <DD Out[3]= 100 200 300 400 0.485 0.490 0.495 0.500 0.505 0.510 0.515 Lets see if we can graphically determine an N that works for e =.05 in the definition of lim s n = 1 2 .
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In[4]:= e = 0.05; p1 = ListPlot @ Table @8 n, s @ n D< , 8 n, 1, 50 <D , DisplayFunction Ø Identity D ; p2 = Plot @8 0.5 - e , 0.5 + e < , 8 x, 0, 50 < , DisplayFunction Ø Identity D ; Show @ p1, p2, AxesOrigin Ø 8 0, 0.5 < , PlotRange Ø 88 0, 50 < , 8 0.3, 0.7 << , DisplayFunction
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This note was uploaded on 11/07/2009 for the course MATH 3224 at Virginia Tech.

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seqexs - Examples of Sequences Example 1: sn = In[1]:= n2 +...

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