Proof
Because we know that for all exists an such that for all we have that, or what is the same
As we can choose such that is positive. So and by the direct comparison test, if converges then so does .
Similarly, so if converges, again by the direct comparison test, so does .
That is both series converge or both series diverge.
Read more about this topic: Limit Comparison Test
Famous quotes containing the word proof:
“War is a beastly business, it is true, but one proof we are human is our ability to learn, even from it, how better to exist.”
—M.F.K. Fisher (1908–1992)
“Ah! I have penetrated to those meadows on the morning of many a first spring day, jumping from hummock to hummock, from willow root to willow root, when the wild river valley and the woods were bathed in so pure and bright a light as would have waked the dead, if they had been slumbering in their graves, as some suppose. There needs no stronger proof of immortality. All things must live in such a light. O Death, where was thy sting? O Grave, where was thy victory, then?”
—Henry David Thoreau (1817–1862)
“O, popular applause! what heart of man
Is proof against thy sweet, seducing charms?”
—William Cowper (1731–1800)