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)
“There is no better proof of a man’s being truly good than his desiring to be constantly under the observation of good men.”
—François, Duc De La Rochefoucauld (1613–1680)
“a meek humble Man of modest sense,
Who preaching peace does practice continence;
Whose pious life’s a proof he does believe,
Mysterious truths, which no Man can conceive.”
—John Wilmot, 2d Earl Of Rochester (1647–1680)