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:
“Distinctions drawn by the mind are not necessarily equivalent to distinctions in reality.”
—Thomas Aquinas (c. 1225–1274)
“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)
“It is commonplace that a problem stated is well on its way to solution, for statement of the nature of a problem signifies that the underlying quality is being transformed into determinate distinctions of terms and relations or has become an object of articulate thought.”
—John Dewey (1859–1952)