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 special case of the Riemann zeta function, $\zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^s}$, where $s=3$.

The value of $\zeta(3)$ is known as Apéry's constant. It is known that $\zeta(3)$ is an irrational number.

However, there is no known closed-form expression for $\zeta(3)$ in terms of elementary functions. Its approximate value is about 1.

2

0

2

0

5

6

9.

3. Final Answer

$\zeta(3)$