The infinite sum of 1/(2^n) and various proofs that this equals 1

Dr.Peterson said:
He's saying essentially what I said here:

That's in answer to your statement that

The sum never "reaches" 1 in the sense that no finite sum of terms (called a partial sum) can equal 1, since the next term would take the sum above 1, which is the limit.

The point is that this is as obvious as you suggest; so why does it concern you that people say it? The statement that the sum "never reaches 1" is perfectly compatible with the fact that the "sum" of all the terms is 1. Do you understand that?
Click to expand...
No, I am really surprised and now very curious how those 2 things can be compatible. I imagine that I am building a bridge. I build the bridge by halving the gap. After an infinite amount of halves, I completely fill the gap. Have I not reached the other side?
 
Steven G said:
You wrote I can't believe that I can't get a straight answer about this. I have read proofs showing that the sum of 1/2^n, as n goes to infinity is exactly 1, and I have read from good sources that it never reaches 1
To me it sounded as if you didn't believe that the sum never reaches 1. I proved that it is the case that no partial sum is ever 1.
Click to expand...
But we have an infinite, not just a partial sum.
 
Mates said:
No, I am really surprised and now very curious how those 2 things can be compatible. I imagine that I am building a bridge. I build the bridge by halving the gap. After an infinite amount of halves, I completely fill the gap. Have I not reached the other side?
Click to expand...
The limit of your bridge building process is the complete bridge. But at no step number N do you reach the other side. Are these 2 statements compatible?
 
Mates said:
No, I am really surprised and now very curious how those 2 things can be compatible. I imagine that I am building a bridge. I build the bridge by halving the gap. After an infinite amount of halves, I completely fill the gap. Have I not reached the other side?
Click to expand...
But that's the point: You can't take an infinite number of steps. That's what they're saying when they say the sum never reaches 1.

You never reach the end in a finite "time"; but the series "reaches" the end in the sense of a limit.

I think you need to go back to the start and tell us what you meant by this:
Mates said:
I can't believe that I can't get a straight answer about this. I have read proofs showing that the sum of 1/2^n, as n goes to infinity is exactly 1, and I have read from good sources that it never reaches 1.
Click to expand...
Mates said:
But I have a feeling they are wrong because some of the better sources I have read have a more rigorous explanation on why it only approaches 1.
Click to expand...
What statements do you agree with, and what do you feel confused about? They are all true, because of what we mean by "sum" in this case.

But maybe also you should quote what was said by the three people you mention, because their actual words may make their meaning clearer.
 
Bridge building. Suppose it takes you exactly 1 minute to build half the remaining bridge at each step. Exactly how minutes, hrs, days, centuries... will it take to finish the bridge?
 
lev888 said:
The limit of your bridge building process is the complete bridge. But at no step number N do you reach the other side. Are these 2 statements compatible?
Click to expand...
Yes, I can understand how those two statements are compatible.
 
But that's the point: You can't take an infinite number of steps. That's what they're saying when they say the sum never reaches 1.

You never reach the end in a finite "time"; but the series "reaches" the end in the sense of a limit.
Click to expand...

But it seems like I can go one step further (pun intended) and say that not only did I build to the limit, but I joined the bridge to the other side. From what I understand, there should be no gap if I really did complete the bridge.

I think you need to go back to the start and tell us what you meant by this:


What statements do you agree with, and what do you feel confused about? They are all true, because of what we mean by "sum" in this case.

But maybe also you should quote what was said by the three people you mention, because their actual words may make their meaning clearer.
Click to expand...

Ok, I will look for it.
 
Steven G said:
Bridge building. Suppose it takes you exactly 1 minute to build half the remaining bridge at each step. Exactly how minutes, hrs, days, centuries... will it take to finish the bridge?
Click to expand...
I would be allowed an infinite number (aleph-null) of minutes, just like I have an aleph-null n.
 
Mates said:
Yes, I can understand how those two statements are compatible.
Click to expand...
Ok, but you don't agree with the following?
"The statement that the sum "never reaches 1" is perfectly compatible with the fact that the "sum" of all the terms is 1"

I would clarify it a bit: "The statement that any finite sum "never reaches 1" is perfectly compatible with the fact that the "sum" of all the terms is 1"

What's the difference between this and the bridge example?
 
lev888 said:
Ok, but you don't agree with the following?
"The statement that the sum "never reaches 1" is perfectly compatible with the fact that the "sum" of all the terms is 1"

I would clarify it a bit: "The statement that any finite sum "never reaches 1" is perfectly compatible with the fact that the "sum" of all the terms is 1"
Click to expand...
That makes perfect sense to me. Maybe this whole misunderstand was just in the wording used.
 
Mates said:
But it seems like I can go one step further (pun intended) and say that not only did I build to the limit, but I joined the bridge to the other side. From what I understand, there should be no gap if I really did complete the bridge.
Click to expand...
Yes, If you really built the bridge, then naturally there will be no gap. The question is if you need an infinite amount of time to build the bridge, then did you really build it? What do you think?
 
Last edited:
Steven G said:
Yes, If you really built the bridge, then naturally there will be no gap. The question is if you need an infinite amount of time to build the, thence you really build it? What do you think?
Click to expand...
I think so. Because, infinity exhausts infinity, right? (assuming equal cardinalities)
 
Mates said:
I think so. Because, infinity exhausts infinity, right? (assuming equal cardinalities)
Click to expand...
How can you possibly build a bridge if it takes an infinite about of time. Will you even be alive from the start until the finish?
 
Steven G said:
How can you possibly build a bridge if it takes an infinite about of time. Will you even be alive from the start until the finish?
Click to expand...
A proper analogy is if I had an infinite amount of time, just like there are an infinite amount of terms as in my OP. Or, I could still build it if I build a piece in decreasing amounts of time such as 1/2^n of an hour or so.
 
Mates said:
A proper analogy is if I had an infinite amount of time, just like there are an infinite amount of terms as in my OP. Or, I could still build it if I build a piece in decreasing amounts of time such as 1/2^n of an hour or so.
Click to expand...
True, but can you really do an infinite number of things in a finite time?
 
Steven G said:
True, but can you really do an infinite number of things in a finite time?
Click to expand...
Do you mean realistically or mathematically? Definitely no realistically, but mathematically I would say yes (based on what I know about all of this).
 
Steven G said:
True, but can you really do an infinite number of things in a finite time?
Click to expand...
Let's say the series in the original question refers to time periods, and each term is the amount of time one "thing" takes. Since the series converges, can we conclude that it takes a finite time to do an infinite number of things?
 
lev888 said:
Let's say the series in the original question refers to time periods, and each term is the amount of time one "thing" takes. Since the series converges, can we conclude that it takes a finite time to do an infinite number of things?
Click to expand...
Yes, for some things!
Suppose you walk at the rate of 1 mile per hour. To walk a mile it will obviously take one hour. Now imagine someone hits a bell after you have 1/2miles left, 1/4 mile left, 1/8 mile left, etc. How many times will this person ring a bell in that 1 hour period?

EDIT: After reading the last few posts I see why you asked me this. I was just challenging the OP with my question in the earlier post.
 
You must log in or register to reply here.
Top