Does 1+2+3+4+... really equal -1/12? Math explained

This is the most misunderstood result in popular mathematics. **The sum $1+2+3+4+\cdots$ DIVERGES to $\infty$ in the usual sense of infinite series.** The series $\sum_{n=1}^{\infty} n$ does NOT converge to any finite value. The $-\frac{1}{12}$ result comes from **analytic continuation of the Riemann zeta function:** $$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$$ This series converges for $\Re(s) > 1$. But the zeta function can be analytically continued to the entire complex plane (except $s=1$). At $s = -1$, we get $\zeta(-1) = -\frac{1}{12}$. **The "Ramanujan sum"** is a different summation method (Ramanujan summation) that assigns values to divergent series in a way consistent with analytic continuation. **Is it "true"?** In the sense of standard convergent series, absolutely not. In the sense of analytic continuation and Ramanujan summation, yes — but these are different definitions of "sum." Physicists use this in Casimir effect calculations and string theory because the analytic continuation gives physically meaningful results, even though the series itself diverges.