Interval of Convergence
When we see the series as a function, we can actually specify an interval for the function so that the series certainly converges over this interval.
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The method is kind of like finding the interval of an ordinary function:
The
term
of series can't be0
, so seta_n ≠ 0
and solve it to get the condition.Take some
convergence tests
, etc.ratio test
.Calculate to get the condition that makes the test passes.
Substitute the
endpoints
of the interval back in the function and see if it also converges.
Example
Solve:
The term can't be zero, so:
And further more, take a
ratio test
which makes it converges:Calculate the
ratio test
to get the interval:Get the interval for
x
:Test the
endpoints
for this interval:In conclusion, the
interval of convergence
is:
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