What blew my mind is this. What is the sum of the infinite series
1, -1, 1, -1, ...
One answer is to look at it like this:
(1 - 1) + (1 - 1) + ... = 0
Another answer is to look at it like this:
1 + (-1 + 1) + (-1 + 1) + ... = 1
But then it gets weirder. What if you add two of the series together like so:
1 + -1 + 1 + -1 + ...
____ 1 + -1 + 1 + ...
(Please ignore the underscores. They’re just there because otherwise Lemmy messes up the whitespace.)
All the terms cancel out except that first 1 again. But this time it’s the sum of two of these series, which means that the sum of one series is 0.5 and somehow not an integer.
The correct answer is that you’re not allowed to add up infinite series like this so that’s why you get contradictory results if you try.
You are actually allowed to add up infinite series like this.
Only that the infinite series have to be convergent, or else you get little of value. The series in your example oscillates forever (and the oscillation distance remains constant), therefore it diverges.
Take the infinite series 1 + 0 + 0 + 0 + ... and add it like you did:
1 + 0 + 0 + 0 + 0 + ... ___ 1 + 0 + 0 + 0 + ...
And you just get 1 + 1 + 0 + 0 + 0 + ... which is just 2 * (1 + 0 + 0 + 0 + ...)
What blew my mind is this. What is the sum of the infinite series
1, -1, 1, -1, ...One answer is to look at it like this:
(1 - 1) + (1 - 1) + ... = 0Another answer is to look at it like this:
1 + (-1 + 1) + (-1 + 1) + ... = 1But then it gets weirder. What if you add two of the series together like so:
1 + -1 + 1 + -1 + ...____ 1 + -1 + 1 + ...(Please ignore the underscores. They’re just there because otherwise Lemmy messes up the whitespace.)
All the terms cancel out except that first 1 again. But this time it’s the sum of two of these series, which means that the sum of one series is 0.5 and somehow not an integer.
The correct answer is that you’re not allowed to add up infinite series like this so that’s why you get contradictory results if you try.
You are actually allowed to add up infinite series like this.
Only that the infinite series have to be convergent, or else you get little of value. The series in your example oscillates forever (and the oscillation distance remains constant), therefore it diverges.
Take the infinite series
1 + 0 + 0 + 0 + ...and add it like you did:1 + 0 + 0 + 0 + 0 + ...___ 1 + 0 + 0 + 0 + ...And you just get
1 + 1 + 0 + 0 + 0 + ...which is just2 * (1 + 0 + 0 + 0 + ...)First step to find 1 + 2 + 3 + … = -1/12
There’s a Wikipedia page about this: https://en.wikipedia.org/wiki/Grandi's_series
The correct answer is that the sum doesn’t have a value, but it you must assign a value to it, then 0.5 is the most correct value.