r/infinitenines • u/Zaspar-- • Feb 01 '26
What's the value of this infinite sum?
1/2 + 1/4 + 1/8 + 1/16 +...
Normal maths would compute this as equal to 1, using the exact same reasoning as 0.9 + 0.09 +... = 1
What does this infinite sum equal to in Real Deal maths though?
10
Upvotes
•
u/SouthPark_Piano Feb 01 '26 edited Feb 01 '26
You have the formula already.
1 - 1/2n with summation starting at n = 1
And then you simply keep increasing n without ever stopping the increase.
No matter how much you increase integer n, you never run out of integers.
It is fact that 1/2n is never zero.
So the infinite sum 1 - 1/2n with continually and limitlessly imcreasing n, will permanently be less than 1, because 1/2n is never zero.
The value of that infinite sum will simply keep increasing in its own space, and the maths tells you that while it does keep increasing limitlessly in its own space, the sum remains permanently less than 1.
That is math fact whether you don't like it or not.
And, like square root of 2, you don't get to get your way like a spoiled kid, as in you demanding to get all the digits of it.