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 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.

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: