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The Casimir effect is a manifestation of $$1+2^3+3^3+\cdots=-\frac{1}{120}.$$

The vacuum energy $E$ in the space between two metal plates, separated by a distance $a$ equals $$E = \frac{ \hbar c \pi^2 }{6a^3}\sum_{n=1}^\infty n^3.$$ This divergent sum is regularized by analytic continuation of the Riemann zeta function, to give $$1+2^s+3^s+\cdots=\zeta(-s)=-\frac{B_{s+1}}{s+1},$$ with $B_s$ a Bernoulli number, hence $\sum_{n=1}^\infty n^3=-\frac{1}{120}.$

The resulting attractive force $-dE/da$ between the metal plates, the Casimir effect, has been demonstrated in experiments, providing one justification for the zeta-function regularization of the divergent sum.

The Casimir effect is a manifestation of $$1+2^3+3^3+\cdots=-\frac{1}{120}.$$

The vacuum energy $E$ in the space between two metal plates, separated by a distance $a$ equals $$E = \frac{ \hbar c \pi^2 }{6a^3}\sum_{n=1}^\infty n^3.$$ This divergent sum is regularized by analytic continuation of the Riemann zeta function, to give $$1+2^s+3^s+\cdots=\zeta(-s)=-\frac{B_{s+1}}{s+1},$$ with $B_s$ a Bernoulli number, hence $\sum_{n=1}^\infty n^3=-\frac{1}{120}.$

The resulting attractive force $-dE/da$ between the metal plates, the Casimir effect, has been demonstrated in experiments, providing one justification for the zeta-function regularization of the divergent sum.

_{In the physics context, what is going on is that the unobservable vacuum energy $E$ is infinite, while the observable force $-dE/da$ is finite. One way to obtain this finite answer is to add a third metal plate, at a distance $L$ from the two plates at separation $a$. The third plate introduces a cutoff in the infinite sum, and the limit $\lim_{L\rightarrow\infty} dE/da$ gives the same finite answer as the zeta-function regularization.}

The Casimir effect is a manifestation of $$1+2^3+3^3+\cdots=-\frac{1}{120}.$$

The vacuum energy $E$ in the space between two metal plates, separated by a distance $a$ equals $$E = \frac{ \hbar c \pi^2 }{6a^3}\sum_{n=1}^\infty n^3.$$ This divergent sum is regularized by analytic continuation of the Riemann zeta function, to give $$1+2^s+3^s+\cdots=\zeta(-s)=-\frac{B_{s+1}}{s+1},$$ with $B_s$ a Bernoulli number, hence $\sum_{n=1}^\infty n^3=-\frac{1}{120}.$

The resulting attractive force $-dE/da$ between the metal plates, the Casimir effect, has been demonstrated in experiments, providing one justification for the zeta-function regularization.

The Casimir effect is a manifestation of $$1+2^3+3^3+\cdots=-\frac{1}{120}.$$

The vacuum energy $E$ in the space between two metal plates, separated by a distance $a$ equals $$E = \frac{ \hbar c \pi^2 }{6a^3}\sum_{n=1}^\infty n^3.$$ This divergent sum is regularized by analytic continuation of the Riemann zeta function, to give $$1+2^s+3^s+\cdots=\zeta(-s)=-\frac{B_{s+1}}{s+1},$$ with $B_s$ a Bernoulli number, hence $\sum_{n=1}^\infty n^3=-\frac{1}{120}.$

The resulting attractive force $-dE/da$ between the metal plates, the Casimir effect, has been demonstrated in experiments, providing one justification for the zeta-function regularization of the divergent sum.

The vacuum energy $E$ in the space between two metal plates, separated by a distance $a$ equals $$E = \frac{ \hbar c \pi^2 }{6a^3}\sum_{n=1}^\infty n^3.$$ This divergent sum is regularized by analytic continuation of the Riemann zeta function, to give $$\sum_{n=1}^\infty n^3=\zeta(-3)=-\frac{1}{120}.$$ The resulting attractive force $-dE/da$ between the metal plates, the Casimir effect, has been demonstrated in experiments, providing one justification for the zeta-function regularization.

The Casimir effect is a manifestation of $$1+2^3+3^3+\cdots=-\frac{1}{120}.$$

The vacuum energy $E$ in the space between two metal plates, separated by a distance $a$ equals $$E = \frac{ \hbar c \pi^2 }{6a^3}\sum_{n=1}^\infty n^3.$$ This divergent sum is regularized by analytic continuation of the Riemann zeta function, to give $$1+2^s+3^s+\cdots=\zeta(-s)=-\frac{B_{s+1}}{s+1},$$ with $B_s$ a Bernoulli number, hence $\sum_{n=1}^\infty n^3=-\frac{1}{120}.$

The resulting attractive force $-dE/da$ between the metal plates, the Casimir effect, has been demonstrated in experiments, providing one justification for the zeta-function regularization.