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)
“...I have come to make distinctions between what I call the academy and literature, the moral equivalents of church and God. The academy may lie, but literature tries to tell the truth.”
—Dorothy Allison (b. 1949)
“Distinctions drawn by the mind are not necessarily equivalent to distinctions in reality.”
—Thomas Aquinas (c. 1225–1274)