It is not true that “no pattern that suggests a value suggests any other”, at least not unless you say more precisely what you are willing to count as a pattern.
Here’s a template describing the pattern you’ve used to argue that 1+2+...=-1/12:
We define numbers aij with the following two properties. First, limj→∞aij=i, so that for each j we can think of (aij) as a sequence that’s looking more and more like (1,2,3,...) as j increases. Second, ∑∞i=1aij=sj where sj→−112, so the sums of these sequences that look more and more like (1,2,3,...) approach −1/12.
(Maybe you mean something more specific by “pattern”. You haven’t actually said what you mean.)
Well, here are some aij to consider. When i>j+1 we’ll let aij=0. When i≤j we’ll let aij=i. And when i=j+1 we’ll let aij=A−(1+⋯+i). Here, A is some fixed number; we can choose it to be anything we like.
This array of numbers satisfies our first property: limj→∞aij=i. Indeed, once j≥i we have aij=i, and the limit of an eventually-constant sequence is the thing it’s eventually constant at.
What about the second property? Well, as you’ll readily see I’ve arranged that for each j we have ∑∞i=1aij=A. So the sequence of sums converges to A.
In other words, this is a “pattern” that makes the sum equal to A. For any value of A we choose.
I believe there are more stringent notions of “pattern”—stronger requirements on how the aij approach i for large j -- for which it is true that every “pattern” that yields a finite sum yields −112. But does this actually end up lower-tech than analytic continuation and the like? I’m not sure it does.
It is not true that “no pattern that suggests a value suggests any other”, at least not unless you say more precisely what you are willing to count as a pattern.
Here’s a template describing the pattern you’ve used to argue that 1+2+...=-1/12:
We define numbers aij with the following two properties. First, limj→∞aij=i, so that for each j we can think of (aij) as a sequence that’s looking more and more like (1,2,3,...) as j increases. Second, ∑∞i=1aij=sj where sj→−112, so the sums of these sequences that look more and more like (1,2,3,...) approach −1/12.
(Maybe you mean something more specific by “pattern”. You haven’t actually said what you mean.)
Well, here are some aij to consider. When i>j+1 we’ll let aij=0. When i≤j we’ll let aij=i. And when i=j+1 we’ll let aij=A−(1+⋯+i). Here, A is some fixed number; we can choose it to be anything we like.
This array of numbers satisfies our first property: limj→∞aij=i. Indeed, once j≥i we have aij=i, and the limit of an eventually-constant sequence is the thing it’s eventually constant at.
What about the second property? Well, as you’ll readily see I’ve arranged that for each j we have ∑∞i=1aij=A. So the sequence of sums converges to A.
In other words, this is a “pattern” that makes the sum equal to A. For any value of A we choose.
I believe there are more stringent notions of “pattern”—stronger requirements on how the aij approach i for large j -- for which it is true that every “pattern” that yields a finite sum yields −112. But does this actually end up lower-tech than analytic continuation and the like? I’m not sure it does.
(One version of the relevant theory is described at https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation.)