Thread: Give Counterexample: b_(n+1)-b_n --> 0, then b_n --> L

given: a_n = b_(n+1) - b_n

[if the limit of a_n = 0, then (b_n) has a limit] = FALSE.

please provide a counterexample. with all my heart i think the converse is true. if have already proven the converse of the converse.

i have no idea where to even begin, since it seems obvious that ... should the difference between consecutive terms approach zero, then the terms approach an agreeable limit...

i.e. i keep coming up with sequences that actually DO converge (e.g. (-1)^n/n ...)

Originally Posted by Skerven

given: a_n = b_(n+1) - b_n

[if the limit of a_n = 0, then (b_n) has a limit] = FALSE.

please provide a counterexample. with all my heart i think the converse is true. if have already proven the converse of the converse.

i have no idea where to even begin, since it seems obvious that ... should the difference between consecutive terms approach zero, then the terms approach an agreeable limit...

i.e. i keep coming up with sequences that actually DO converge (e.g. (-1)^n/n ...)

if you're assuming that L is "finite", then a counter-example would be the converse is obviously true.

Another is , where .

for .
By the way, any function satisfying (these imply that tends to asymptotically) with (this implies diverges to ) can be your sequence .
Draw such a function on a paper and try to see what I mean...

Originally Posted by bkarpuz

By the way, any function satisfying (these imply that tends to asymptotically) with (this implies diverges to ) can be your sequence .

Yeah, as strange as it seems, I get it. f(x) -> Inf while f'(x) -> 0. Thanks for that generality.
@redsoxfan325: I also thought of that odd series.