$\int_\mathbb Z 3^{-|x|}d\zeta(x)$

Question

Let $\zeta$ be the counting measure on $(\mathbb Z,\mathscr P(\mathbb Z))$. Calculate $\int_\mathbb Z 3^{-|x|}d\zeta(x)$. How can I calculate this integral?

Answer 1

Notice that

$$I=\int_\mathbb{Z}3^{-\lvert x\rvert}\,\mathrm{d}\zeta(x)=\sum_{n\in\mathbb{Z}}3^{-\lvert n\rvert}.$$

We split the sum up and get that

$$I=\sum_{n\in\mathbb{Z}^-}3^{-\lvert n\rvert}+\sum_{n\in\mathbb{Z}^+}3^{-\lvert n\rvert}+3^{-\lvert 0\rvert}=1+2\sum_{n=1}^\infty\frac{1}{3^n}.$$

This is just a convergent geometric series, so we get that

$$I=1+2\cdot\frac{1}{2}=2.$$