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Science & Alternative Science / Infinity
« on: June 24, 2014, 07:55:35 AM »
I might as well post this on the forum where I still know some of the people.
Analytic continuation allows us to approximate an infinite series as a function. The more terms you add to the series, the more accurately you describe the function. So at the limit, when you have all the terms, that series can be considered equal to that function.
This allows us to assign meaningful numerical values to infinite series, even divergent ones.
For example, the infinite series "1 + 2 + 3 + 4 + 5... " can be evaluated. However, the result is disturbingly counter-intuitive. If you stop adding terms at any finite point, you'll have a larger and larger number as the result. However, if you "evaluate this after an infinite number of terms, you'll find the sum to actually be -1/12.
There are a variety of proofs for this, and this number is demonstrably a meaningful . This result is actually seen in physics. Furthermore this result is a foundation in string theory for the number of required dimensions.
Analytic continuation allows us to approximate an infinite series as a function. The more terms you add to the series, the more accurately you describe the function. So at the limit, when you have all the terms, that series can be considered equal to that function.
This allows us to assign meaningful numerical values to infinite series, even divergent ones.
For example, the infinite series "1 + 2 + 3 + 4 + 5... " can be evaluated. However, the result is disturbingly counter-intuitive. If you stop adding terms at any finite point, you'll have a larger and larger number as the result. However, if you "evaluate this after an infinite number of terms, you'll find the sum to actually be -1/12.
There are a variety of proofs for this, and this number is demonstrably a meaningful . This result is actually seen in physics. Furthermore this result is a foundation in string theory for the number of required dimensions.