**Problem** We want to find the exact value of $\zeta(2)$ where $\zeta$ is the [[Riemann Zeta Function]]. > [!NOTE] > We have $n^{2} \geq n(n-1)$ so $\sum_{n=1}^{m} \frac{1}{n^{2}} \leq 1+ \sum_{n=1}^{m-1} \frac{1}{n(n+1)} \leq 1+ \sum_{n=1}^{\infty} \frac{1}{n(n+1)} = 2.$Hence the series converges since [[Series with Non-Negative Terms Converges Iff Partial Sums Are Bounded Above]] (we know that the [[Telescoping Sum|telescoping sequence]], $\sum_{n=1}^{\infty} \frac{1}{n(n+1)} =1$). > **Claim** We have $\sum_{n=1}^{\infty} \frac{1}{n^{2}} = \frac{\pi^{2}}{6}$ **Proof** ...