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

[Mates]

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.

Here is one such "proof":

Let x = 1/2+1/4+1/8+1/16...
Multiply both sides of the equation by 2: 2(1/2+1/4+1/8+1/16...) = 2x.
We see that the first term becomes 1 plus the same infinite series.
1+1/2+1/4+1/8...=2x So we can sub in x, since we let x equal the same infinite series. 1+x=2x
x = 1

And I know of 2 more "proofs" if anyone wants to know them. 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.

You can try to look this up on Wiki, but I warn you, their answer is very very confusing https://en.wikipedia.org/wiki/Series_(mathematics)

If you know, please help!

 

[lev888]

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.

Here is one such "proof":

Let x = 1/2+1/4+1/8+1/16...
Multiply both sides of the equation by 2: 2(1/2+1/4+1/8+1/16...) = 2x.
We see that the first term becomes 1 plus the same infinite series.
1+1/2+1/4+1/8...=2x So we can sub in x, since we let x equal the same infinite series. 1+x=2x
x = 1

And I know of 2 more "proofs" if anyone wants to know them. 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.

You can try to look this up on Wiki, but I warn you, their answer is very very confusing https://en.wikipedia.org/wiki/Series_(mathematics)

If you know, please help!

No members of the partial sum sequence reach 1. But the limit of the sequence is 1.
Is it true that 0.9999... = 1.0? Same idea.

 

[khansaheb]

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.

Here is one such "proof":

Let x = 1/2+1/4+1/8+1/16...
Multiply both sides of the equation by 2: 2(1/2+1/4+1/8+1/16...) = 2x.
We see that the first term becomes 1 plus the same infinite series.
1+1/2+1/4+1/8...=2x So we can sub in x, since we let x equal the same infinite series. 1+x=2x
x = 1

And I know of 2 more "proofs" if anyone wants to know them. 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.

You can try to look this up on Wiki, but I warn you, their answer is very very confusing https://en.wikipedia.org/wiki/Series_(mathematics)

If you know, please help!

Have you studied Zeno's paradox?

 

[Mates]

No members of the partial sum sequence reach 1. But the limit of the sequence is 1.
Is it true that 0.9999... = 1.0? Same idea.

0.9999 ... does equal 1. There is a proof of that.

 

[Mates]

Have you studied Zeno's paradox?

Yes, but I do not see how it helps.

 

[Dr.Peterson]

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.
...
And I know of 2 more "proofs" if anyone wants to know them. 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.

The partial sums (sum of the first 1, 2, 3, ... terms) are 1/2, 3/4, 7/8, 15/16, ... .

No one of those is 1; but what we call the "sum" of the infinite series is not one of those, but the number they approach as a limit, which is 1.

In fact, the nth partial sum is 1 - 1/2^n, which clearly approaches 1 as n increases.

So all the statements you make are true: the "sum", which is defined as a limit, is 1; the finite partial sums never reach 1 (in fact, if they did, then the limit would be greater than 1!); the partial sums approach 1.

Similarly, the value of 0.999... is defined as the limit of partial sums 0.9, 0.99, 0.999, ..., none of which is 1, but they approach 1 as a limit.

 

[Mates]

The partial sums (sum of the first 1, 2, 3, ... terms) are 1/2, 3/4, 7/8, 15/16, ... .

No one of those is 1; but what we call the "sum" of the infinite series is not one of those, but the number they approach as a limit, which is 1.

In fact, the nth partial sum is 1 - 1/2^n, which clearly approaches 1 as n increases.

So all the statements you make are true: the "sum", which is defined as a limit, is 1; the finite partial sums never reach 1 (in fact, if they did, then the limit would be greater than 1!); the partial sums approach 1.

Similarly, the value of 0.999... is defined as the limit of partial sums 0.9, 0.99, 0.999, ..., none of which is 1, but they approach 1 as a limit.

I think I just found an indisputable proof that the sum not only has a limit of 1 or defined to be 1, but equals exactly 1. What do you think of this geometric proof? (At first the demonstration wasn't obvious, but now I realize that the decreasing rectangles alone must reach the bottom right corner of the unit square (of area 1). And similarly, the decreasing squares must reach the bottom right of the unit square. There is no space left over for the ratios to fill.)

edit: for some reason I cannot post a youtube video.

Here is the same idea on Wiki https://en.wikipedia.org/wiki/1/4_+_1/16_+_1/64_+_1/256_+_⋯

 

[lev888]

0.9999 ... does equal 1. There is a proof of that.

Yes, and there is a proof that your sum is 1. But you had a feeling that it is not 1. I am giving you another sum to illustrate the same idea.
0.999.. = 0.9 + 0.09 + 0.009 + ... = 1

 

[Dr.Peterson]

I think I just found an indisputable proof that the sum not only has a limit of 1 or defined to be 1, but equals exactly 1.

You've missed the main point: we define an infinite sum as the limit, so these three statements all mean the same thing!

(You can't actually add infinitely many numbers, so such a definition is necessary.)

 

[Mates]

Yes, and there is a proof that your sum is 1. But you had a feeling that it is not 1. I am giving you another sum to illustrate the same idea.
0.999.. = 0.9 + 0.09 + 0.009 + ... = 1

Oh ok, thanks!

 

[Mates]

You've missed the main point: we define an infinite sum as the limit, so these three statements all mean the same thing!

(You can't actually add infinitely many numbers, so such a definition is necessary.)

Is this how it always is with infinite sums, or if not, why does this have to be a definition and not other infinite sums? For example, is the infinite sum of 1/k! = e also a definition?

 

[Dr.Peterson]

Is this how it always is with infinite sums, or if not, why does this have to be a definition and not other infinite sums? For example, is the infinite sum of 1/k! = e also a definition?

What I said was that we define an infinite sum as a limit of partial sums -- not "this" sum, but any sum. This is about the whole concept -- you can't literally add any infinite list of numbers, so any such sum has to be defined as a limit.

Each sum, of course, is a different limit (if the limit exists at all).

 

[Mates]

What I said was that we define an infinite sum as a limit of partial sums -- not "this" sum, but any sum. This is about the whole concept -- you can't literally add any infinite list of numbers, so any such sum has to be defined as a limit.

Each sum, of course, is a different limit (if the limit exists at all).

Okay, thanks a lot!

 

[Steven G]

Of course, if 1/2+1/4+1/8+1/16... =1, then no partial sum can equal 1. Since all the terms are positive, if the first n terms sum to 1, then the remaining (infinite) terms will be positive--if not infinite. Then the infinite sum will be greater than 1 which is not true.

 

[khansaheb]

Yes, but I do not see how it helps.

In particular, study

Achilles and the Tortoise​

 

[Mates]

Of course, if 1/2+1/4+1/8+1/16... =1, then no partial sum can equal 1. Since all the terms are positive, if the first n terms sum to 1, then the remaining (infinite) terms will be positive--if not infinite. Then the infinite sum will be greater than 1 which is not true.

I don't understand what you are saying. How can the first n terms equal 1? Doesn't it take an infinite number of terms to equal 1? n can't equal infinity.

 

[lev888]

I don't understand what you are saying. How can the first n terms equal 1? Doesn't it take an infinite number of terms to equal 1? n can't equal infinity.

Steven used a proof by contradiction to prove that a partial sum can't equal 1.

 

[Mates]

Steven used a proof by contradiction to prove that a partial sum can't equal 1.

I know that a partial sum of 1 can't equal 1. What does that have to do with whether or not all infinite terms equals 1 or not?

 

[Dr.Peterson]

I don't understand what you are saying. How can the first n terms equal 1? Doesn't it take an infinite number of terms to equal 1? n can't equal infinity.

He's saying essentially what I said here:

So all the statements you make are true: the "sum", which is defined as a limit, is 1; the finite partial sums never reach 1 (in fact, if they did, then the limit would be greater than 1!); the partial sums approach 1.

That's in answer to your statement that

I have read from good sources that it never reaches 1.

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?

 

[Steven G]

I don't understand what you are saying. How can the first n terms equal 1? Doesn't it take an infinite number of terms to equal 1? n can't equal infinity.

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.