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Exactly 1

An infinite series can describe a number exactly, like

 


Explanation

Without making a calculation we can see that the result cannot exceed 1. We add each time only half the previous term. But is the result really exactly 1, or just a little bit smaller than 1 and does it only approach 1?
You can double a term, and immediately subtract it again. Then you obtain the original value again, because

2 apples – 1 apple = 1 apple

Apply this scheme to the series, so

then it gives

and this is again

But look carefully now. One term has disappeared, as is missing here. So is this just hocus pocus? Let us look to this calculation from the other side, and write it as

Here we also add each time only half of the previous term. Now we are sure the number 1 is correct, then this was our starting value. You can read this formula from left to the right and from right to the left, so our calculation is correct, and

 


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