Interval of Convergence
Last updated
Last updated
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 be 0, so set a_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.
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:
