- ...interval
- For the interested reader, some remarks on
endpoints are contained in footnotes throughout.
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- ...x=-2
- It is not hard to see
that the series diverges in each of these cases by other
means. For example, when x=2 we are looking at the sum
22#22. Since the terms of this sum
do not tend to 0, the sum will diverge. The case of x=-2
is similar.
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- ...determined
- The series converges at both 26#26.
The method for showing this is through what is called the
integral test, a comparison between the sum and an
improper integral.
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- ...x=2
- It turns out that the series
converges at x=2 but that it diverges at x=0.
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- ...above
- With a few technical improvements, the
arguments presented here really work for
any power series.
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- ...applies
- To be honest,
there is one case for which this discussion is not quite
right.
When the radius of convergence is infinite,
the series converges even faster than this: the power series
converges faster than any geometric series.
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