You can look for the insight here.

In short, it we introduce a non-Archimedean numerical system that interprets generalized summations of infinite series (in Ramanujan's sense or Zeta function regularization) as "standard part" of the sum of the series, then in this system the quantity of all natural numbers $\omega_-$ has the standard part $-1/2$ and the quantity of non-negative integers $\omega_+$ is greater by 1 (for zero), so has the standard part $1/2$.

Consequently, from Faulhaber's formula for Ramanujan's summation,

$$\operatorname{st}\omega_-^n=B_n$$

$$\operatorname{st}\omega_+^n=B^*_n$$

where Where $B_n$ are the first Bernoulli numbers and $B^∗_n$ are the second Bernoulli numbers.

So the Bernoulli numbers are the standard part of the powers of the quantity of naturals.

Similarly,

$$\operatorname{st} e^{z\omega_-}=\frac{z}{e^{z}-1}$$

and

$$\operatorname{st} e^{-z\omega_-}=\frac{z}{1-e^{-z}}$$

You can look for the insight here.

In short, it we introduce a non-Archimedean numerical system that interprets generalized summations of infinite series (in Ramanujan's sense or Zeta function regularization) as "standard part" of the sum of the series, then in this system the quantity of all natural numbers $\omega_-$ has the standard part $-1/2$ and the quantity of non-negative integers $\omega_+$ is greater by 1 (for zero), so has the standard part $1/2$.

Consequently, from Faulhaber's formula for Ramanujan's summation,

$$\operatorname{st}\omega_-^n=B_n$$

$$\operatorname{st}\omega_+^n=B^*_n$$

where Where $B_n$ are the first Bernoulli numbers and $B^∗_n$ are the second Bernoulli numbers.

So the Bernoulli numbers are the standard part of the powers of the quantity of naturals.

Similarly,

$$\operatorname{st} e^{z\omega_-}=\frac{z}{e^{z}-1}$$

and

$$\operatorname{st} e^{-z\omega_-}=\frac{z}{1-e^{-z}}$$