The problem asks to evaluate the infinite sum $\sum_{k=1}^{\infty} \frac{1}{k^3}$. This is Apéry's constant.

1. Problem Description

The problem asks to evaluate the infinite sum $\sum_{k=1}^{\infty} \frac{1}{k^3}$. This is Apéry's constant.

2. Solution Steps

The sum $\sum_{k=1}^{\infty} \frac{1}{k^3}$ is known as Apéry's constant. It is a mathematical constant that appears in many areas of mathematics, including number theory and mathematical physics. It is denoted by $\zeta(3)$, where $\zeta(s)$ is the Riemann zeta function.

The Riemann zeta function is defined as:

$\zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^s}$

The value of Apéry's constant is approximately 1.202056903159594285399738161511449990764986292...

It is known that $\zeta(2) = \frac{\pi^2}{6}$. However, there is no known closed-form expression for $\zeta(3)$ in terms of elementary functions such as polynomials, exponentials, logarithms, and trigonometric functions. Apéry proved that $\zeta(3)$ is irrational.

Although there is no simple closed-form expression for $\zeta(3)$, its value is known to a high degree of accuracy.

3. Final Answer

$\zeta(3)$