Abel's theorem

In mathematics, Abel's theorem for power series relates a limit of a power series to the sum of its coefficients. It is named after Norwegian mathematician Niels Henrik Abel. Let the Taylor series be a power series with real coefficients with radius of convergence Suppose that the series converges. Then is continuous from the left at that is, The same theorem holds for complex power series provided that entirely within a single Stolz sector, that is, a region of the open unit disk where for some fixed finite . Without this restriction, the limit may fail to exist: for example, the power series converges to at but is unbounded near any point of the form so the value at is not the limit as tends to 1 in the whole open disk. Note that is continuous on the real closed interval for by virtue of the uniform convergence of the series on compact subsets of the disk of convergence. Abel's theorem allows us to say more, namely that is continuous on The Stolz sector has explicit formulaand is plotted on the right for various values. The left end of the sector is , and the right end is . On the right end, it becomes a cone with angle , where . As an immediate consequence of this theorem, if is any nonzero complex number for which the series converges, then it follows that in which the limit is taken from below. The theorem can also be generalized to account for sums which diverge to infinity. If then However, if the series is only known to be divergent, but for reasons other than diverging to infinity, then the claim of the theorem may fail: take, for example, the power series for At the series is equal to but We also remark the theorem holds for radii of convergence other than : let be a power series with radius of convergence and suppose the series converges at Then is continuous from the left at that is, The utility of Abel's theorem is that it allows us to find the limit of a power series as its argument (that is, ) approaches from below, even in cases where the radius of convergence, of the power series is equal to and we cannot be sure whether the limit should be finite or not.

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