Why is the sum of an infinite geometric series of fractions = zero?

HallsofIvy

MHF Helper
Apr 2005
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7,909
The closest you can approach that limit of 2 meters is just up to, but never quite at the 2-meter limit.
Then you still do not understand the limit concept! A sequence can be "approaching" a number but the limit IS that number.
 
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HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
I agree. But without substituting zero for $r^{n+1}\ \ $, the equation of $x = \dfrac{1}{1-r}\ \ $ makes no sense.
No, the equation $x= \lim_{n\to\infty}\frac{1- r^{n+1}}{1- r}$ does make sense without "substituting zero for $r^{n+1}\ \ $".
 
Aug 2007
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It's even more disappointing in that this is the second time I'm doing this. 50 years ago I took mathematics up through Calculus 2 at university. 3 years ago I decided I didn't really understand anything I learned back then. Corporate management duties, even in the software industry, don't call on using Lagrange transformations all that much. My priorities in school were more about learning how to do problems over why they are done that way.

Over the past 3 years, I've re-taken Algebra I $ II, Geometry, Trigonometry and now Pre-Calculus with the focus on why things are done the way they are, and where and how the ideas originated. It was a Eureka moment for me to discover how Pythagoras actually figured out his eponymous formula. I don't learn formulas anymore without working through their derivations. It became very time-consuming to derive some of the Trig identities. I find I can't move forward in my studies if I don't thoroughly understand why each step is done as it is. Seeming inconsistencies are major stumbling blocks for me.

All this is to say thank you. Clearly, 14 years of mathematics, plus another 3 of review, didn't cover this topic in enough depth. I'll dig deeper into real numbers. If you have any recommendations for me, I would appreciate it.
If 0.99999.... isn't EQUAL TO one (1), can you tell me how far from one (1) it is?