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Example Question #4 : Concepts Of Convergence And Divergence
Which of following intervals of convergence cannot exist?
For any , the interval for some
For any such that , the interval
cannot be an interval of convergence because a theorem states that a radius has to be either nonzero and finite, or infinite (which would imply that it has interval of convergence ). Thus, can never be an interval of convergence.
Example Question #1 : Radius And Interval Of Convergence Of Power Series
Find the interval of convergence of for the series .
Using the root test,
Because 0 is always less than 1, the root test shows that the series converges for any value of x.
Therefore, the interval of convergence is:
Example Question #2 : Radius And Interval Of Convergence Of Power Series
Find the interval of convergence for of the Taylor Series .
Using the root test
herefore, the series only converges when it is equal to zero.
This occurs when x=5.