The Instigator
baggins
Pro (for)
Losing
0 Points
The Contender
bencbartlett
Con (against)
Winning
24 Points

Two is equal to One

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Post Voting Period
The voting period for this debate has ended.
after 8 votes the winner is...
bencbartlett
Voting Style: Open Point System: 7 Point
Started: 9/2/2012 Category: Science
Updated: 4 years ago Status: Post Voting Period
Viewed: 1,562 times Debate No: 25411
Debate Rounds (3)
Comments (5)
Votes (8)

 

baggins

Pro

In R2 I will present a proof that 2 = 1.

Please don't worry. The proof is flawed. In R2 it will be task of Con to spot the flaw. If she is able to do this, she wins the debate. If she is unable to spot the flaw in R2, I win this debate.

If Con is unable to spot the flaw in R2 (or provides an errenous explanation), I will provide the correct explanation in R3. Con need not do anything in R1 and R3.

I will be using Taylor series expansion and logarithms. Nothing in much detail. But you need to be comfortable with basics of these two concepts and series calcuations to accept this debate.

Hoping to get a sincere opponent.
bencbartlett

Con

I accept this challenge - it'll be nice to actually be able to argue mathematics on here for once, since I've tried starting quite a few math debates before. I'll post my solution as soon as I find an error in whatever proof you offer, and I commit to working this problem alone. Good luck!
Debate Round No. 1
baggins

Pro

Thanking bencbartlett for accepting this debate. I agree that it feels nice to debate mathematics.

I am numbering the steps in order to assist discussion.

Taylor expansion of ln (1+x) is:
1-> ln (1+x) = x - x^2/2 + x^3/3 - x^4/4 + x^5/5 -+-

Putting x = 1.
2-> ln (1 + 1) = ln (2) = 1 - 1/2 + 1/3 - 1/4 + 1/5 -+-

Rearranging the terms
3-> ln (2) = (1 + 1/3 + 1/5 +++) - (1/2 + 1/4 + 1/6 +++)

4-> Let P = 1 + 1/3 + 1/5 +++

5-> Let Q = 1/2 + 1/4 + 1/6 +++

From steps 3, 4 and 5
6-> ln (2) = P - Q

Consider Q
7-> Q = 1/2 + 1/4 + 1/6 + 1/8 + 1/10 +++

Taking 1/2 as common from LHS.
8-> Q = 1/2 [1 + 1/2 + 1/3 + 1/4 + 1/5 +++]

Rearranging terms
9-> Q = 1/2 [ (1 + 1/3 + 1/5 +++) + (1/2 + 1/4 + 1/6 +++) ]

Subsituting series in step 9 with help of step 4 and 5,
10-> Q = 1/2 [ P + Q]

11-> 2Q = P + Q

12-> Q = P

Substituting Q in step 6.
13-> ln (2) = P - P

14-> ln (2) = 0

Taking antilogs on both sides
15-> 2 = 1

Best of Luck to my esteemed opponent.
bencbartlett

Con

My opponent’s argument breaks down at step 2. The Taylor expansion for ln(x) has a radius of convergence of 1, around the point x = 1, such that 0<x<2 (as opposed to 0≤x≤2) as I will show below, meaning that the Taylor expansion of ln(x-1) likewise has a radius of convergence of 1 around the point x = 0, such that -1<x<1. By inserting x=1 into the Taylor expansion for ln(x-1), he is causing the series to become divergent, thereby inducing the error in the proof that appears to make 2=1.

Proof that the Taylor expansion for ln(x+1) fails at x=1:

1) Differentiating ln(1+x), we have:

d/dx (ln(1+x)) = 1/(1+x).

2) Trivially, 1/(1+x) = 1/(1-(-x)) = 1/(1-y), where y≡-x.

3) Taking the Maclaurin expansion of 1/(1-y), we have:

1/(1-y) = 1+y+y^2+y^3+y^4+…

4) Note that the result in step 3 is the expression of a geometric series, which has limits of evaluation at |y|<1. Since y≡-x, |y|<1 is equivalent to |-x|<1. Substituting 1 for x, we have |-1|=1<1, which is false, since 1=1 (again, note the difference between 1<1 and 1≤1). For example, substituting x=1 into the previous expression yields:

1/(1+x) = 1/(1-y) = 1+(-1)+1+(-1)+1+(-1)+…

This expression is divergent, resulting in the error that causes 2 to appear to equal 1. This concludes the proof.

****************************************************************************

I greatly enjoyed this debate; you’ll have to challenge me to another some time!

Debate Round No. 2
baggins

Pro

My opponent is correct that the series is divergent. As a result many algebric operations I have performed are not valid.

Since my opponent has spotted the mistake, I will concede. Thanks to him for debating this topic with me.
bencbartlett

Con

Thank you as well for offering the topic - I enjoyed it!
Debate Round No. 3
5 comments have been posted on this debate. Showing 1 through 5 records.
Posted by baggins 4 years ago
baggins
Changed the time limit.
Posted by Ore_Ele 4 years ago
Ore_Ele
There is not a problem to working alone.
Posted by baggins 4 years ago
baggins
In 72 hours, you will be able to ask people in mathematics department in some university and get response from them. If you commit to working alone, I will change time limit to 72 hours.

Any flaw which invalidates the proof is sufficient.
Posted by Ore_Ele 4 years ago
Ore_Ele
Also, I'd take this but I need the 72 hours per round.
Posted by Ore_Ele 4 years ago
Ore_Ele
So we only need to point out a mathematical flaw, or does it have to be the specific one that you are looking for?
8 votes have been placed for this debate. Showing 1 through 8 records.
Vote Placed by Jessalyn 4 years ago
Jessalyn
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Total points awarded:03 
Reasons for voting decision: I think it's obvious...Con won. But HUGE props to Pro, too, for all the time spent on this debate. I admire the mathematical skill and interest.
Vote Placed by Ore_Ele 4 years ago
Ore_Ele
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Reasons for voting decision: Con solved the puzzle and Pro conceded.
Vote Placed by famer 4 years ago
famer
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Reasons for voting decision: Pro conceded
Vote Placed by Zaradi 4 years ago
Zaradi
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Reasons for voting decision: I'm going to pretend like I understood what the fvck was going on and just give con args because pro conceded....
Vote Placed by vmpire321 4 years ago
vmpire321
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Reasons for voting decision: uhhh...
Vote Placed by socialpinko 4 years ago
socialpinko
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Reasons for voting decision: I'm glad there wasn't contention regarding what the flaw was at the end of the debate and that Pro conceded because honestly, I was utterly confused for the entirety of the debate.
Vote Placed by AlwaysMoreThanYou 4 years ago
AlwaysMoreThanYou
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Reasons for voting decision: derp
Vote Placed by Microsuck 4 years ago
Microsuck
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Reasons for voting decision: con spotted the flaw.