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where lim sup denotes the limit superior, the limit as n approaches infinity of the supremum of the sequence values after the nth position. If the sequence values are unbounded so that the lim sup is ?, then the power series does not converge near a, while if the lim sup is 0 then the radius of convergence is ?, meaning that the series converges on the entire plane.
Without loss of generality assume that . We will show first that the power series converges for , and then that it diverges for .
First suppose . Let not be or
For any , there exists only a finite number of such that .
Now for all but a finite number of , so the series converges if . This proves the first part.
Conversely, for , for infinitely many , so if , we see that the series cannot converge because its nth term does not tend to 0.
Theorem for several complex variables
Let be a multi-index (a n-tuple of integers) with , then converges with radius of convergence (which is also a multi-index) if and only if