August 27th, 2015, 06:14 PM  #1 
Senior Member Joined: Jan 2014 From: The backwoods of Northern Ontario Posts: 390 Thanks: 70  Proof that an Infinite Series Adds to 1
We will prove that: $\displaystyle \frac{1}{2}+\frac{1}{4}+\frac{1}{8}... = 1$ (exactly) Proof: (1) Let n = $\displaystyle \frac{1}{2}+\frac{1}{4}+\frac{1}{8}...$ Multiply each side of (1) by 2 (2) 2n = $\displaystyle 1 + \frac{1}{2}+\frac{1}{4}+\frac{1}{8}...$ Subtract each side of line (1) from each side of line (2) (3) n = 1 One can also use this technique to prove that .999... is exactly equal to 1. 
August 27th, 2015, 11:04 PM  #2 
Math Team Joined: Nov 2014 From: Australia Posts: 689 Thanks: 244 
Be careful when using proofs like this. Saying "Let $n = \dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{8} + ...$" implies that the series converges, which may not be the case. There are (false) proofs that the series $1  1 + 1  1 + ...$ converges to $0$, $1$ and $\dfrac{1}{2}$ using this principle, but the series does not converge at all. If you want to formalise this proof, you have to first prove that the series converges. 
August 28th, 2015, 06:21 AM  #3 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
This is, in fact, a "geometric series". A geometric series is any series of the form $\displaystyle \sum_{n=0}^\infty ar^n= a+ ar+ ar^2+ ar^3+ \cdot\cdot\cdot$. One can prove that a geometric series converges if and only if r< 1 and then converges to $\displaystyle \frac{a}{1 r^2}$. Here, we can write the series as $\displaystyle \frac{1}{2}\left(1+ \frac{1}{2}+ \frac{1/4}+ \cdot\cdot\cdot\right)$ so 1/2 times a geometric sequence with a= 1 and r= 1/2. Since 1/2< 1, this converges to $\displaystyle \frac{1}{2}\frac{1}{1 \frac{1}{2}}= \frac{1}{2}\frac{2}{2 1}= 1$. (This is also a geometric series with a= 1/2 and r= 1/2 so converges to $\displaystyle \frac{\frac{1}{2}}{1 \frac{1}{2}}= \frac{\frac{1}{2}}{\frac{1}{2}}= 1$) (It can also be thought of as a geometric series with a= 1 and r= 1/2 with the first "a" term missing. The geometric series with a= 1 and r= 1/2, $\displaystyle 1+ \frac{1}{2}+ \frac{1}{4}+ \frac{1}{8}+ \cdot\cdot\cdot$ converges to $\displaystyle \frac{1}{1 \frac{1}{2}}= 2$ and subtracting the missing first term, "1" gives a sum of 1.) 
August 28th, 2015, 06:42 AM  #4 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,664 Thanks: 2644 Math Focus: Mainly analysis and algebra  
August 28th, 2015, 11:41 AM  #5 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
How in the world did that "2" sneak in there?! Yes, of course. Thank you.

August 28th, 2015, 11:49 AM  #6 
Senior Member Joined: Jan 2014 From: The backwoods of Northern Ontario Posts: 390 Thanks: 70 
Perhaps I'm stupid, but I see no necessity of first proving that the series converges. Whether it converges or not, the proof I provided is purely algebraic: Multiplying each side of an equation by 2, and Subtracting the same thing from each side of an equation. 
August 28th, 2015, 12:32 PM  #7  
Math Team Joined: May 2013 From: The Astral plane Posts: 2,203 Thanks: 901 Math Focus: Wibbly wobbly timeywimey stuff.  Quote:
For example: S = 1 + 2 + 3 + 4 + ... 2S = 2 + 4 + 6 + 8 + .... S = 2S  S = 1  3  5  7  ... But can we say that 1 + 2 + 3 + 4 + ... = 1  3  5  7  .... ? Dan  
August 28th, 2015, 12:44 PM  #8  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,664 Thanks: 2644 Math Focus: Mainly analysis and algebra  Quote:
Suppose that $$s = 1  1 + 1  1 + \cdots$$Then $$s = 1 + 1  1 + 1 \cdots = 1 + s$$Thus $$2s = 1 \implies \boxed{s = \tfrac12}$$ This uses all the same techniques that yours does, but is patently not true. The partial sums never get closer to $\frac12$ at all! What you are really saying is that "I get the right answer, so why should I prove that the method is sound". But that means that what you have is not a proof that the sum is what you say it is because applied to another sequence we don't know whether the result is correct or not. We have to check by another method. Last edited by v8archie; August 28th, 2015 at 12:48 PM.  
August 28th, 2015, 02:30 PM  #9 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,664 Thanks: 2644 Math Focus: Mainly analysis and algebra  
August 28th, 2015, 03:51 PM  #10  
Senior Member Joined: Jan 2014 From: The backwoods of Northern Ontario Posts: 390 Thanks: 70  Quote:
2S= 2 + 4 + 6 + 8 + ... S= 1 + 2 + 3 + 4 + ... Subtract both sides and you have: S = 1 + 2 + 3 + 4 + ...  

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