Suppose that a sequence (xn) satisfies the following inequality:
abs(xn+1 - xn) < abs(xn - xn-1) for n = 2,3,4,.....
Does this sequence converge or is there a counterexample for it being divergent? If it converges, how would you prove it?
I would love some help on this particular problem. I'm not really sure if it converges or not. Thanks.
Hello, PvtBillPilgrim!
Suppose that a sequence satisfies the following inequality:
. . . for
Does this sequence converge or is there a counterexample for it being divergent?
If it converges, how would you prove it?
TPH is absolutely correct.
. . Let me translate it . . .
It says: the difference of consecutive terms is decreasing.
. . The terms are "getting closer."
It would seem that the series converges.
But the Harmonic Series: . . satisfies the inequality,
. . and it diverges.
We have found a counterexample.
Therefore, the sequence may or may not converge.