Distinctions
A series can be uniformly convergent and absolutely convergent without being uniformly absolutely-convergent. For example, if ƒn(x) = xn/n on the open interval (−1,0), then the series Σfn(x) converges uniformly by comparison of the partial sums to those of Σ(−1)n/n, and the series Σ|fn(x)| converges absolutely at each point by the geometric series test, but Σ|fn(x)| does not converge uniformly. Intuitively, this is because the absolute-convergence gets slower and slower as x approaches −1, where convergence holds but absolute convergence fails.
Read more about this topic: Uniform Absolute-convergence
Famous quotes containing the word distinctions:
“Mankind are an incorrigible race. Give them but bugbears and idols—it is all that they ask; the distinctions of right and wrong, of truth and falsehood, of good and evil, are worse than indifferent to them.”
—William Hazlitt (1778–1830)
“Television ... helps blur the distinction between framed and unframed reality. Whereas going to the movies necessarily entails leaving one’s ordinary surroundings, soap operas are in fact spatially inseparable from the rest of one’s life. In homes where television is on most of the time, they are also temporally integrated into one’s “real” life and, unlike the experience of going out in the evening to see a show, may not even interrupt its regular flow.”
—Eviatar Zerubavel, U.S. sociologist, educator. The Fine Line: Making Distinctions in Everyday Life, ch. 5, University of Chicago Press (1991)
“Again we have here two distinctions that are no distinctions, but made to seem so by terms invented by I know not whom to cover ignorance, and blind the understanding of the reader: for it cannot be conceived that there is any liberty greater, than for a man to do what he will.”
—Thomas Hobbes (1579–1688)