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:
“The thing with Catholicism, the same as all religions, is that it teaches what should be, which seems rather incorrect. This is “what should be.” Now, if you’re taught to live up to a “what should be” that never existed—only an occult superstition, no proof of this “should be”Mthen you can sit on a jury and indict easily, you can cast the first stone, you can burn Adolf Eichmann, like that!”
—Lenny Bruce (1925–1966)
“Sculpture and painting are very justly called liberal arts; a lively and strong imagination, together with a just observation, being absolutely necessary to excel in either; which, in my opinion, is by no means the case of music, though called a liberal art, and now in Italy placed even above the other two—a proof of the decline of that country.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)
“He who has never failed somewhere, that man can not be great. Failure is the true test of greatness. And if it be said, that continual success is a proof that a man wisely knows his powers,—it is only to be added, that, in that case, he knows them to be small.”
—Herman Melville (1819–1891)