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I didn't understand the proof for 0.5 + 0.25 + ... = 1 itself. By his visual number line analogy, wouldn't 2/3 + (2/3)^2 + ... = 1 too? Visually, that too seems

I didn't understand the proof for 0.5 + 0.25 + ... = 1 itself. By his visual number line analogy, wouldn't 2/3 + (2/3)^2 + ... = 1 too? Visually, that too seems to "approach" 1.

evanb 10y ago
It approaches 2. 2/3+(2/3)^2 = 2/3 + 4/9 = 6/9+4/9=10/9 > 1, and all the terms are positive, so as you add more terms it will get farther from 1. If you do 10 terms you get to 1.96532somethingsomething. If you do 100 terms you get 1.[seventeen 9s]508somethingsomething.
- BillBohan 10y ago
  leereves is correct. The sequence is 2/3 + 2/9 + 2/27 + 2/81 + 2/243 + ... and it does approach 1 In your second term you should take 2/3 of the remaining 1/3, not (2/3)^2
Houshalter 10y ago
I like the algebraic approach: x = 1/2 + 1/2*1/2 + 1/2*1/2*1/2 + ... 2x = 1 + 1/2 + 1/2*1/2 + 1/2*1/2*1/2 + ... 2x = 1 + x x = 1
leereeves 10y ago
The number line analogy does work for 2/3, in the same way it worked for 9/10. It represents the repeating trinary number 0.22222... = 1. But the analogy only corresponds to p + p^2 + p^3 ... = 1 when p = 1/2 because that's the only time the remainder = p. For 2/3, the number line analogy instead gives 2/3 + 1/3 * 2/3 + (1/3)^2 * 2/3 ...
ssivark 10y ago
Coming from someone who is "used to" this idea for a long time, I'm not sure how helpful my explanation would be to a learner, but I decided to take a shot at it, just in case it helps someone. That said, in my experience, a significant part of "understanding" involves getting used to things and expanding the scope of what you consider "natural". -- Suppose you wish to walk from 0 to 1. You'll have to walk half the distance. Then half of the remaining half. Then half of the remaining quarter. Then half of the remaining eighth. If you keep doing this "asymptotically", you will get very very close to 1 and (almost) reach there. That's the basic idea. This is also known as Zeno's paradox: https://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Paradoxes_of_motion https://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Paradoxes_o... Instead of halving step sizes on each iteration, you could reduced by a different fraction. In general, this is called a geometric series: https://en.wikipedia.org/wiki/Geometric_series https://en.wikipedia.org/wiki/Geometric_series When the ratio is (2/3), after two steps you overshoot 1. (You can see that (2/3) = 0.66, (2/3)^2 = 0.44 )

I didn't understand the proof for 0.5 + 0.25 + ... = 1 itself. By his visual number line analogy, wouldn't 2/3 + (2/3)^2 + ... = 1 too? Visually, that too seems to "approach" 1.

evanb 10y ago

It approaches 2. 2/3+(2/3)^2 = 2/3 + 4/9 = 6/9+4/9=10/9 > 1, and all the terms are positive, so as you add more terms it will get farther from 1. If you do 10 terms you get to 1.96532somethingsomething. If you do 100 terms you get 1.[seventeen 9s]508somethingsomething.

BillBohan 10y ago
leereves is correct. The sequence is 2/3 + 2/9 + 2/27 + 2/81 + 2/243 + ... and it does approach 1 In your second term you should take 2/3 of the remaining 1/3, not (2/3)^2

Houshalter 10y ago

I like the algebraic approach: x = 1/2 + 1/2*1/2 + 1/2*1/2*1/2 + ... 2x = 1 + 1/2 + 1/2*1/2 + 1/2*1/2*1/2 + ... 2x = 1 + x x = 1

leereeves 10y ago

The number line analogy does work for 2/3, in the same way it worked for 9/10. It represents the repeating trinary number 0.22222... = 1. But the analogy only corresponds to p + p^2 + p^3 ... = 1 when p = 1/2 because that's the only time the remainder = p. For 2/3, the number line analogy instead gives 2/3 + 1/3 * 2/3 + (1/3)^2 * 2/3 ...

ssivark 10y ago

Coming from someone who is "used to" this idea for a long time, I'm not sure how helpful my explanation would be to a learner, but I decided to take a shot at it, just in case it helps someone. That said, in my experience, a significant part of "understanding" involves getting used to things and expanding the scope of what you consider "natural". -- Suppose you wish to walk from 0 to 1. You'll have to walk half the distance. Then half of the remaining half. Then half of the remaining quarter. Then half of the remaining eighth. If you keep doing this "asymptotically", you will get very very close to 1 and (almost) reach there. That's the basic idea. This is also known as Zeno's paradox: https://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Paradoxes_of_motion https://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Paradoxes_o... Instead of halving step sizes on each iteration, you could reduced by a different fraction. In general, this is called a geometric series: https://en.wikipedia.org/wiki/Geometric_series https://en.wikipedia.org/wiki/Geometric_series When the ratio is (2/3), after two steps you overshoot 1. (You can see that (2/3) = 0.66, (2/3)^2 = 0.44 )