October 3rd, 2016, 01:45 AM  #21  
Senior Member Joined: Apr 2014 From: Glasgow Posts: 1,837 Thanks: 592 Math Focus: Physics, mathematical modelling, numerical and computational solutions  Quote:
$\displaystyle 0.5000... = 0.5$ $\displaystyle 0.4999... = 0.4\dot{9} = 0.5$ So the two values you quoted are equal to each other. I think you're trying to say something like this: Define two functions: $\displaystyle f(n) = \frac{2}{5} + \sum_{i=1}^{n} \frac{9}{10^i} $ $\displaystyle g(n) = 0.5$ As $\displaystyle n \rightarrow \infty$, $\displaystyle g(n)  f(n) \rightarrow 0$. That's fine. However, if you set n to be infinity, what happens? $\displaystyle g(\infty)  f(\infty) = 0$ But then you tell me that n cannot be equal to infinity. Why not?  
October 3rd, 2016, 08:46 AM  #22  
Senior Member Joined: Mar 2015 From: New Jersey Posts: 861 Thanks: 68  Quote:
2) infinity is not a number. You conveniently ignored my following post: $\displaystyle \lim_{n\rightarrow \infty}$1/n =0 but 1/n never equals zero. Suppose you listed the nplace decimals in [0,1) in order: .0000000 .0000001 .0000002 .............. As n approaches $\displaystyle \infty$, they approach a countably infinite list of the real numbers. If you let n= $\displaystyle \infty$, they would all be zero by your thinking, because .00000.....1 = 0 when n = $\displaystyle \infty$ .00000.....2 = 0 when n = $\displaystyle \infty$ ............. $\displaystyle \lim_{n\rightarrow \infty}$.000....1 = 0, but .000....1 never equals 0. Last edited by skipjack; October 4th, 2016 at 03:04 PM.  
October 3rd, 2016, 09:12 AM  #23 
Global Moderator Joined: Dec 2006 Posts: 16,369 Thanks: 1172  
October 3rd, 2016, 09:38 AM  #24  
Math Team Joined: Dec 2013 From: Colombia Posts: 6,391 Thanks: 2100 Math Focus: Mainly analysis and algebra  This is correct. $\frac25+\sum \limits_{n=2}^\infty \frac9{10^n}=0.5$ does not depend on evaluating the sum at $n=\infty$, but it does represent the value of the infinite decimal. Quote:
The number of digits cannot "approach infinity". It is either finite or it is not. The length of the decimal is not convergent so it doesn't converge to $\infty$. Indeed the phrase "approaches infinity" in the context of limits has a precisely defined meaning that does not use the concept of "infinity". Your attempt to use the phrase as in some way related to decimals of infinite length is an inaccurate blurring of precise definitions. Last edited by skipjack; October 4th, 2016 at 03:05 PM.  
October 4th, 2016, 03:29 AM  #25  
Senior Member Joined: Apr 2014 From: Glasgow Posts: 1,837 Thanks: 592 Math Focus: Physics, mathematical modelling, numerical and computational solutions  I'm not trying to prove anything, I'm just quoting the result. What assumptions am I making that prevent me from putting infinity as a limit of a summand? If you mean this... Quote:
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As CRGreathouse argued in previous threads, if you want to mess around with your own number system, that's fine, but if you do that you are no longer talking about the reals or the decimal number system. You can even create your own system that includes weird things like 0.000...1 or $\displaystyle 0.\dot{0}5$ or whatever, but those numbers are certainly not decimal numbers. Last edited by skipjack; October 4th, 2016 at 03:08 PM.  
October 4th, 2016, 04:56 AM  #26 
Senior Member Joined: Apr 2014 From: Glasgow Posts: 1,837 Thanks: 592 Math Focus: Physics, mathematical modelling, numerical and computational solutions  
October 4th, 2016, 07:40 AM  #27  
Global Moderator Joined: Dec 2006 Posts: 16,369 Thanks: 1172  Quote:
They wouldn't all be zero. The last in the list for any value of n is the decimal containing n nines, which approaches 1 as n tends to $\infty$.  
October 4th, 2016, 08:06 AM  #28 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 861 Thanks: 68 
The nplace decimals in [0,1) are unique, rational and countably finite. As n $\displaystyle \rightarrow \infty$ they remain unique, and become real and countably infinite. 
October 4th, 2016, 08:22 AM  #29 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 861 Thanks: 68 
.00....01 and .00....02 are not the same no matter how many decimal places you take. .49999.... and .50000.. are not the same no matter how many decimal places you take. 
October 4th, 2016, 08:28 AM  #30 
Senior Member Joined: Apr 2014 From: Glasgow Posts: 1,837 Thanks: 592 Math Focus: Physics, mathematical modelling, numerical and computational solutions  

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