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[1]/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.

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