Do you know Leibniz convergence criterium?
As a lemma to proving the "alternating series test" using the Cauchy criterion, I am told I need to show the following, (and then use this fact to prove the AST). (I am told it should be no more than a sentence or two)
Lemma. Let be a positive monotonically decreasing sequence. Suppose that for all , implies that . Then satisfies the Cauchy criterion
(A series is said to satsify the cauchy crtierion if and only if for all , there exists such that imply
NOTE: I am not asking for the actual proof of the AST. Just this basic fact above which is a prerequisite to the proof.
Thanks for any help,
No sir I do not. Again, my goal is not to prove the actual AST. I am just trying to justify a preliminary assumption. Perhaps this will make what I want a little clearer.
Statement 1. For all , implies
Statement 2. For all , there exists such that implies
I need to show that Statement 1. implies Statement 2. (Prove that Statement 1. is sufficient for Statement 2.). My teacher explained that it should be very short.