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

[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.

 

[jaedaliu]

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.

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.

 

[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.

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

 

[chuckywang]

Originally posted by: Syringer
Or more mathematically:

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



I can't think of one at all

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

 

[Whitecloak]

sin(x) or cosine(x) with the range 0<=x<90 converges to 0 or 1 but never reaches the end points

 

[chuckywang]

Originally posted by: Whitecloak
sin(x) or cosine(x) with the range 0<=x<90 converges to 0 or 1 but never reaches the end points

That's not a sequence.

 

[QED]

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.

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|

 

[George P Burdell]

We're NOT gonna do your homework for you

 

[Syringer]

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.

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

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.

1/n converges to 0?

 

[chuckywang]

Originally posted by: Syringer

Originally posted by: chuckywang


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

1/n converges to 0?

Yeah, 1/n converges to 0. My statement still stands.

 

[IAteYourMother]

i thought 1/n converged to 0

 

[chuckywang]

Originally posted by: IAteYourMother
i thought 1/n converged to 0

It does.

 

[IAteYourMother]

Originally posted by: chuckywang

Originally posted by: IAteYourMother
i thought 1/n converged to 0

It does.

yeah, nvm.

 

[Safeway]

1/n ... Series ... 1/1 + 1/2 + 1/3 + 1/4 ... seems like it converges to 2

 

[chuckywang]

Originally posted by: Safeway
1/n ... Series ... 1/1 + 1/2 + 1/3 + 1/4 ... seems like it converges to 2

The harmonic series diverges actually.

 

[IAteYourMother]

Originally posted by: chuckywang

Originally posted by: Safeway
1/n ... Series ... 1/1 + 1/2 + 1/3 + 1/4 ... seems like it converges to 2

The harmonic series diverges actually.

yep, in a series of 1/n^p, p>1 for it to converge... i think

 

[chuckywang]

Originally posted by: IAteYourMother

Originally posted by: chuckywang

Originally posted by: Safeway
1/n ... Series ... 1/1 + 1/2 + 1/3 + 1/4 ... seems like it converges to 2

The harmonic series diverges actually.

yep, in a series of 1/n^p, p>1 for it to converge... i think

Yep, the good 'ole zeta function.