The problem is to find the sum of the infinite series $\sum_{k=1}^{\infty} \frac{1}{k^3}$.

1. Problem Description

The problem is to find the sum of the infinite series $\sum_{k=1}^{\infty} \frac{1}{k^3}$.

2. Solution Steps

The given series is $\sum_{k=1}^{\infty} \frac{1}{k^3}$. This is a p-series with $p=3$.

A p-series is defined as

$\sum_{k=1}^{\infty} \frac{1}{k^p}$

The p-series converges if $p>1$ and diverges if $p \le 1$.

In this case, $p=3$, so the series converges.

The series $\sum_{k=1}^{\infty} \frac{1}{k^3}$ is a specific case of the Riemann zeta function, defined as $\zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^s}$.

Therefore, we have $\sum_{k=1}^{\infty} \frac{1}{k^3} = \zeta(3)$.

The value of $\zeta(3)$ is Apéry's constant, which is an irrational number. There is no known closed-form expression for $\zeta(3)$ in terms of other known constants like $\pi$ or $e$. Its approximate value is 1.202056903159594285399738161511449990764986292...

Thus, the sum of the series is $\zeta(3)$.

3. Final Answer

$\zeta(3)$