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:
“O, popular applause! what heart of man
Is proof against thy sweet, seducing charms?”
—William Cowper (1731–1800)
“From whichever angle one looks at it, the application of racial theories remains a striking proof of the lowered demands of public opinion upon the purity of critical judgment.”
—Johan Huizinga (1872–1945)
“The fact that several men were able to become infatuated with that latrine is truly the proof of the decline of the men of this century.”
—Charles Baudelaire (1821–1867)