In order to prove that 0.9999~ (that's 0.9 repeating) is equal to 1 we first need to come up with an equation that will allow 9 to be repeated after the decimal as much as we want. The best way to do this would to add up 0.9 + 0.09 + 0.009 + ... and so on. The only way to express this is with a summation. The summation is shown below, and is also simplified down to an easier to read equation:
Now that we have a simplified equation, let's see it in action. We'll start with n=1 and work our way up from there:
As you can see, the large the valuer of n, the more times 9 repeats after the decimal. So what is the best way to get 9 to repeat forever? The best way is to make n = infinity. This can be expressed with the simple limit below:
That limit could be easily rewritten as:
As you can see, 10 to the power of infinity becomes infinity. 1 divided by infinity is 0. That simplifies the whole equation down to 1 - 0, which is just 1. Thus proving that 0.999~ is equal to 1.
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