Outline of Proof
After subtracting a constant from, we may assume that . Let . Then substituting and performing a simple manipulation of the series results in
Given, pick n large enough so that for all and note that
when z lies within the given Stoltz angle. Whenever z is sufficiently close to 1 we have
so that when z is both sufficiently close to 1 and within the Stoltz angle.
Read more about this topic: Abel's Theorem
Famous quotes containing the words outline of, outline and/or proof:
“One by one objects are defined—
It quickens: clarity, outline of leaf
But now the stark dignity of
entrance—Still, the profound change
has come upon them: rooted, they
grip down and begin to awaken”
—William Carlos Williams (1883–1963)
“It is the business of thought to define things, to find the boundaries; thought, indeed, is a ceaseless process of definition. It is the business of Art to give things shape. Anyone who takes no delight in the firm outline of an object, or in its essential character, has no artistic sense.... He cannot even be nourished by Art. Like Ephraim, he feeds upon the East wind, which has no boundaries.”
—Vance Palmer (1885–1959)
“There are some persons in this world, who, unable to give better proof of being wise, take a strange delight in showing what they think they have sagaciously read in mankind by uncharitable suspicions of them.”
—Herman Melville (1819–1891)