## Wednesday, November 15, 2017

### [wvaejjpf] A collection of divergent sums

Riemann zeta function:
zeta(0) = 1+1+1+1+... = -1/2
zeta(-1) = 1+2+3+4+... = -1/12
zeta(-2) = 1+4+9+16+... = 0
zeta(-1/2) = sqrt(1)+sqrt(2)+sqrt(3)+sqrt(4)+... ~= -0.2078862
zeta(1/2) = 1/sqrt(1)+1/sqrt(2)+1/sqrt(3)+1/sqrt(4)+... ~= -1.4603545
zeta(1) = 1/1+1/2+1/3+1/4+... diverges (harmonic series)
zeta(2) = 1/1+1/4+1/9+1/16+... = pi^2/6 (not a divergent series, but still cool)

Dirichlet eta function:
eta(0) = 1-1+1-1+... = 1/2 (Grandi's series)
eta(-1) = 1-2+3-4+... = 1/4
eta(-2) = 1-4+9-16+... = 0
eta(-1/2) = sqrt(1)-sqrt(2)+sqrt(3)-sqrt(4)+... ~= 0.3801048
eta(1) = 1/1-1/2+1/3-1/4+... = ln 2 (not a divergent series, but still cool)

Geometric series:
1+2+4+8+... = -1
1-2+4-8+... = 1/3
1+10+100+1000+... = -1/9
1-10+100-1000+... = 1/11

Series involving factorials:
0!-1!+2!-3!+4!-5!+... ~= 0.5963474 (Wikipedia)
0!+1!+2!+3!+4!+... = ExpIntegralEi/E ~= 0.6971749 (infinite sum of factorials) (unsure about this one, derived the analytic continuation myself, have not seen published elsewhere)

Although using analytic continuation to obtain values is a legitimate real thing (well, it's actually complex, ha!), saying a divergent series "equals" a certain value is kind of a mathematical in-joke, understandable only if you understand analytic continuation.