Partial sums of 1+1/4+1/9+... converging to π²/6
Partial sums approach π²/6 ≈ 1.6449 slowly. Euler proved the limit equals π²/6 in 1734, connecting analysis to geometry.
The first eight terms of the Basel series: 1/n^2
Each term 1/n^2 decreases rapidly. Their sum converges to exactly pi^2/6 ~1.6449.
Euler's proof idea: sin(x)/x as an infinite product
sin(x)/x = (1−x²/π²)(1−x²/4π²)(1−x²/9π²)…
Comparing x² koefisien: −1/π² − 1/4π² − 1/9π² − … = −1/6
Therefore 1/1² + 1/2² + 1/3² + … = π²/6 ∎
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