Number theory question

[Steven G]

Notation: Let {x} = the fractional part of x. For example {45.67} = .67

Suppose you have a sequence s_n that approaches infinity. Then how can you find {s_n} as n approaches infinity? I doubt that the limit equals infinity + .25.

There is a video from Professor Michael Penn that shows that the limit as n approaches infinity of {(2+sqrt(2))^2} = 1. You can view the video here.

 

[lev888]

If we don't know anything about the sequence other than the limit is infinity, how can we make any conclusions about the fractional part?

 

[Steven G]

If we don't know anything about the sequence other than the limit is infinity, how can we make any conclusions about the fractional part?

How can there be a fractional part at infinity? If for example ithe limit of s_n at infinity was 5.7, then I can understand that the fractional part of the limit is .7.

 

[lev888]

How can there be a fractional part at infinity? If for example ithe limit of s_n at infinity was 5.7, then I can understand that the fractional part of the limit is .7.

`n-(1/n)` approaches infinity and the fractional part approaches 1.

 

[Dr.Peterson]

... the limit as n approaches infinity of {(2+sqrt(2))^2} = 1. You can view the video here.

How does that make any sense? There's no n there!

Apparently you meant the limit as n approaches infinity of {(2+sqrt(2))^n} = 1.

Suppose you have a sequence sn that approaches infinity. Then how can you find {sn} as n approaches infinity? I doubt that the limit equals infinity + .25.

Doesn't the video answer your question?

Obviously the limit of the fractional part can't be greater than 1, since fractional parts are never greater than 1. I would imagine that not all such limits will exist, but there's no reason to think his answer is wrong.

In fact, here are the first few powers, which make the answer obvious:

3.414213562
11.65685425
39.79898987
135.882251
463.9310242
1583.959595
5407.976331
18463.98614
63039.99188
215231.9952
734847.9972
2508927.998
8566015.999

 

[Steven G]

How does that make any sense? There's no n there!

Apparently you meant the limit as n approaches infinity of {(2+sqrt(2))^n} = 1.

Doesn't the video answer your question?

Obviously the limit of the fractional part can't be greater than 1, since fractional parts are never greater than 1. I would imagine that not all such limits will exist, but there's no reason to think his answer is wrong.

In fact, here are the first few powers, which make the answer obvious:

3.414213562
11.65685425
39.79898987
135.882251
463.9310242
1583.959595
5407.976331
18463.98614
63039.99188
215231.9952
734847.9972
2508927.998
8566015.999

Yes, I left out the power n. OK, your list made it clear. You're an excellent tutor. Thank you.
I understood every line in the video but just had a problem with the answer being correct. It is so bizarre to me that a limit can approach infinity with a fractional part approaching a constant. I just could not visualize that! But now I can!

 

[Steven G]

`n-(1/n)` approaches infinity and the fractional part approaches 1.

This was a perfect example. Excellent job. Combined with this example and Dr P's list I understand how this can happen. This is an amazing fact that I will never forgot. Wow. Sequences can have limits on both ends. Amazing!

 

[Dr.Peterson]

An interesting illustration of the fact that an example, either concrete or theoretical, can communicate better than a logical argument what is really going on.

What's interesting to me, which I don't think was pointed out, is that the fractional part can never be 1; but the limit is 1.

 

[JeffM]

Isn't this simply Zeno's paradox in another guise?

 

[Steven G]

What's interesting to me, which I don't think was pointed out, is that the fractional part can never be 1; but the limit is 1.

The fact that the limit was 1 only troubled me for a couple of seconds as I realized that .9999.... = 1
I am completed ok now with a limit approaching oo while the fractional part approaches some value in [0, 1]

This forum really is great, isn't it?

 

[topsquark]

The fact that the limit was 1 only troubled me for a couple of seconds as I realized that .9999.... = 1

Now you've done it. You said that 0.9999.... = 1. This is going to draw the trolls like a magnet!

-Dan

 

[Deleted member 4993]

Now you've done it. You said that 0.9999.... = 1. This is going to draw the trolls like a magnet!

-Dan

No - No- I don't think so. Trolls live under the bridge - and they only count bridge traffic.

Infinity is a bridge too far....

 

[Harry_the_cat]

Interesting! Can someone give an example where the limit of {x} is not 1?

Also, do you say that a limit approaches infinity? Or do you say that there is no limit?

 

[Harry_the_cat]

I suppose {(2+sqrt(2))^n +0.7}=0.7. I think I answered my own first question!

 

[Steven G]

I suppose {(2+sqrt(2))^n +0.7}=0.7. I think I answered my own first question!

There is always a 1st time. I worry more about the last time I answer my own question.

 

[Steven G]

Now you've done it. You said that 0.9999.... = 1. This is going to draw the trolls like a magnet!
-Dan

I always tell non-believers that I will give in if they produce just one number between .99... and 1.

 

[Steven G]

An interesting illustration of the fact that an example, either concrete or theoretical, can communicate better than a logical argument what is really going on.

The proof given by Penn was not at all hard to follow. In fact, it was easy to follow. I just could not believe that if the limit was infinity that the fractional part could even exist. However after seeing your list and lev's amazing example it all became crystal clear.
What happened here, as you are pointing out, is the opposite of what usually happens. Usually when I see a proof of a fact (that I can follow) it clears up any doubt in my mind that it is true and I know that it will work with numerical examples. What happened here is the complete opposite.

The fact that I did not look at the sequence on my own that you posted really shook me up. When confronted with something I don't believe I need to experiment more. Just looking at the sequence you provided was enough to convince me. I should have done that on my own. Now I never would have come up with the sequence supplied by lev, it really made everything clear to me and I now can come up with such an example next time.

 

[Cubist]

It follows from the binomial expansion that...

[math] \left( 2+\sqrt{2} \right)^n = a+b\sqrt{2}[/math]
...where a and b are integers. But interestingly, as n increases then a/b seems to tend towards √2. Plugging in numbers...

       a          b        a/b
======================================
        2          1  2.0
        6          4  1.5
       20         14  1.42857142857142
       68         48  1.41666666666666
      232        164  1.41463414634146
      792        560  1.41428571428571
     2704       1912  1.41422594142259
     9232       6528  1.41421568627450
    31520      22288  1.41421392677674
   107616      76096  1.41421362489486
   367424     259808  1.41421357310013
  1254464     887040  1.41421356421356
  4283008    3028544  1.41421356268886
 14623104   10340096  1.41421356242727
 49926400   35303296  1.41421356238239
170459392  120532992  1.41421356237468
581984768  411525376  1.41421356237336
     ...and sqrt(2) = 1.41421356237309

I can't think of an intuitive reason for this, and I wonder if it's related to the original problem or just coincidence.

BTW: There's a nice recursive formula for successive values of a,b. I'll post details if anyone is interested.