Evaluate the infinite product Π($n=1$ to ∞) (1 - 1/n²) and prove its convergence.

Step 1: Rewrite the general term

First, we rewrite the general term of the product by finding a common denominator and then factoring the numerator. This step simplifies the expression, making it easier to identify cancellations in the product.

Step 2: Write out the partial product

Next, we write out the partial product $P_N$ by substituting the rewritten general term. This allows us to observe the pattern of terms and identify which terms will cancel out.

Step 3: Rearrange and cancel terms

We rearrange the terms in the numerator and denominator to group similar factors. This makes the cancellation process more apparent. Notice how many terms will cancel out between the numerator and denominator.

Step 4: Perform cancellations

After canceling common factors from the numerator and denominator, we are left with a much simpler expression for the partial product $P_N$. The terms $2, 3, \dots, N-1$ from the first part of the numerator cancel with $2, 3, \dots, N-1$ from the first part of the denominator. Similarly, $3, 4, \dots, N$ from the second part of the numerator cancel with $3, 4, \dots, N$ from the second part of the denominator.

Step 5: Evaluate the limit of the partial product

To find the value of the infinite product, we take the limit of the partial product $P_N$ as $N$ approaches infinity. Dividing both the numerator and denominator by $N$ simplifies the limit calculation, showing that the product converges to $1/2$.

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