A while ago, there was a great hype about the “identity”
∞∑n=1n=−112.
Apart from some series manipulations where the validity seems to be at least questionable, the derivation of this always goes through the zeta function:
Where the series converges, the zeta function is defined by
ζ(s)=∞∑n=11ns
and outside that range by analytic continuation. And it turns out that inserting s=−1 formally results in
ζ(−1)=−112=∞∑n=1n
However looking at the series in isolation, there is no indication that the zeta function should be chosen.
An obvious way to get an analytic function that at one point gives the sum of all natural numbers is
f(x)=∞∑n=1nxn
at x=1, however (not surprisingly) that function diverges at 1.
Therefore my question:
Is it possible to get another finite value for the series by analytic continuation of another series?
Concretely, do there exist continuous functions f1,f2,f3,… such that
On some non-empty open subset S of C, f(x)=∑∞n=1fn(x) converges to an analytic function.
At some point x0, fn(x0)=n for all positive integers n.
The analytic continuation of f is well defined and finite at x0.
f(x0)≠−1/12
What if we demand the functions fn to be analytic rather than just continuous?
Answer
What if we demand the functions fn to be analytic rather than just continuous?
No problem. Define
fn(s)=n(n−(−1)n)s,
where ks is as usual defined using the real value of logk (works since n−(−1)n>0). Then fn(0)=n for all n, and by a standard argument the series converges absolutely and locally uniformly for Res>2. We compute
∞∑n=1fn(s)=∞∑n=1n(n−(−1)n)s=∞∑n=1(1(n−(−1)n)s−1+(−1)n(n−(−1)n)s)=∞∑n=11(n−(−1)n)s−1+∞∑n=1(−1)n(n−(−1)n)s=(12s−1+11s−1+14s−1+13s−1+…)+(−12s+11s−14s+13s−16s+15s−…)=∞∑m=11ms−1+∞∑m=1(−1)m−1ms=ζ(s−1)+η(s)
for Res>2. This has an analytic continuation to C∖{2}, and the value at 0 is
ζ(−1)+η(0)=−112+12=512.
One can by similar means obtain different values.
Such summation methods are however very ad-hoc, as far as I know every "reasonable" summation method assigns either +∞ (the natural value) or −112 to the divergent series. I admit that I don't know a good definition of "reasonable" for summation methods (except maybe "extends 'limit of partial sums', is linear and stable", but that definition excludes several widely used summation methods).
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