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YAMT: Sequence that gets closer and closer to a number but does not converge?

S

Syringer

Lifer
Or more mathematically:

|a_(n+1) - a| < |a_n - a|

😕

I can't think of one at all 🙁

Also, for bonus points, can you find a sequence that converges to "a" but does NOT get "closer and closer", that is, violates that property above.
 
Originally posted by: Syringer
Or more mathematically:

|a_(n+1) - a| < |a_n - a|

😕

I can't think of one at all 🙁

Also, for bonus points, can you find a sequence that converges to "a" but does NOT get "closer and closer", that is, violates that property above.
Click to expand...

for the bonus: isn't the definition given a convergence?

I'd find an answer to the first point, but I"m too lazy to pull out my old textbooks. I think that the answer lies in a sequence that alternates between converging to -2 and 2 in an alternate fashion. That should satisfy all points that you stated.
 
Originally posted by: Syringer
Or more mathematically:

|a_(n+1) - a| < |a_n - a|

😕

I can't think of one at all 🙁

Also, for bonus points, can you find a sequence that converges to "a" but does NOT get "closer and closer", that is, violates that property above.
Click to expand...

The sequence of all 2's converges to 2, but doesn't get closer.
 
Originally posted by: Syringer
Or more mathematically:

|a_(n+1) - a| < |a_n - a|

😕

I can't think of one at all 🙁
Click to expand...

The sequence 1/n, where n is a positive integer gets closer and closer to -1, but does not converge to -1.
 
Originally posted by: Syringer

Also, for bonus points, can you find a sequence that converges to "a" but does NOT get "closer and closer", that is, violates that property above.
Click to expand...


Let A_n = 1/n for odd n, and
= 1/n^2 for even n.

This forms the series 1, 1/4, 1/3, 1/16, 1/5, 1/36,1/7 etc. which converges to a=0.
Yet for each even n, we actually have | a_(n+1) - a | > |a_n -a|
 
Originally posted by: chuckywang
Originally posted by: Syringer
Or more mathematically:

|a_(n+1) - a| < |a_n - a|

😕

I can't think of one at all 🙁

Also, for bonus points, can you find a sequence that converges to "a" but does NOT get "closer and closer", that is, violates that property above.
Click to expand...

The sequence of all 2's converges to 2, but doesn't get closer.
Click to expand...

Yeahh I ended up doing that 🙂

The sequence 1/n, where n is a positive integer gets closer and closer to -1, but does not converge to -1.
Click to expand...

1/n converges to 0? 😕
 
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