What's this mean?

Question

http://upload.wikimedia.org/math/7/c/d/7cd33d9a37abfc25e33d0cdb2256366c.png

Answer 1

I assume you know what the “1 –“ part means, as well as the “= 0”. You are probably asking about the weird symbol with the 9/10^i.

The notation is used to denote a series of summations of the number 9/10^i where i starts at 1 and keeps going to m.
You could also write the same thing as,
9/10^1 + 9/10^2 + 9/10^3 + 9/10^4 + ….. + 9/10^m
is the same as,
.9 + .09 + .009 + .0009 + .... + 0
You are just adding up all the numbers between the i = 1 term and the i = m term.
The number on bottom tells you where to start and the number on top tells you where to stop.

As you can see, as i gets very large, 9/10^i gets very small (since the denominator is growing very fast). The smaller 9/10^i becomes, the less its adds to the overall value. As i goes to infinity, 10^i becomes infinite and thus 9/10^i becomes zero…so it does not add anything to the summation.

When this series of additions gets put into the formula, they are saying that the series, essentially, equals 1 (since 1 – 1 = 0). But they leave the variable m in the series…they don’t give you a finite number. They are saying that there is some number, m, that makes that statement true. After m number of 9s after the decimal place (.99999999...999), the series equals one.