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Exactly 1

An infinite series can express a number exactly, like

 


Explanation

Even without a calculation you can see that the result cannot be greater than 1. Every time we only add half the previous term. But is the result exactly 1, or a bit smaller than 1 and does it only approach 1?

1/2 1/4 1/8 ···

Let's take a look from the other side to this calculation. We start with 1, split that in two halfs, and write the tautology

We split one of these halfs further in two, and so on

This can continue forever. We're now sure that 1 is correct, because that was our initial value. You can read the formula from left-to-right and from right-to-left.

 


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