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Cell["\<\
To what does the sequence a[n] = 1 + (-1)^n / n^2 converge? The \
answer leaps out of a graph if we plot the points
 ( -1/n, a[n]  ).\
\>", "Text",
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Cell["\<\
a[n_] = 1.+ (-1)^n / (n^2);
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Cell["\<\
Looks like the points cluster near ( 0, 1).  The limit of the \
sequence is the  y-coordinate of the limit point, which is  1.  To illustrate \
the epsilon-tolerances, we zoom in on the graph, and add two lines for  1 + \
epsilon and  1 - epsilon.  In this case, epsilon is  .004.\.1b\.1bThe \
\"tail\" lies between the lines.\
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a[n_] = 1.+ (-1)^n / (n^2);
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plot2 =Plot[ {1.004,.996},{x,-.15,0} ];
Show[ plot1, plot2 ];\
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Another more \"advanced\" concept is the idea of a limit superior.  \
Some sequences don't settle down to a limit but instead careen about wildly. \
The following graph illustrates a sequence that has a limit superior of  2.5. \
 That is, no matter which tail we look at, we can always find a value that is \
very close to  2.5, but we cannot do it for any larger y-value (e.g., not for \
2.6).\.1b\
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