Question

Prove that
(1-1/2²)(1-1/3²) …(1-1/n²)=n+1/2 n
for all n ⩾ 2.

Answer 1

To prove the product (1 - 1/2²)(1 - 1/3²) …(1 - 1/n²) = (n+1)/(2n), one can show the pattern of cancellation in the fraction series, which simplifies down to the desired result. This relies on the telescoping nature of the products.

Step-by-step explanation:

The student has asked to prove that the product (1 - 1/2²)(1 - 1/3²) …(1 - 1/n²) is equal to (n+1)/(2n) for all n ≥ 2.

To prove this, we can observe the general term in the product:

• The nth term is of the form (1 - 1/n²).
• This can be rewritten as (n² - 1)/(n²) or ((n - 1)(n + 1))/(n²).

We notice a pattern of cancellation when we consider the product sequentially. For example:

• For the second term, (1 - 1/3²), it simplifies to (2· 3)/(3²) = 2/3.
• Pairing this with the previous term (1 - 1/2²), which simplifies to 3/4, we see 2/3 cancels out a part of 3/4.

Continuing this process, we end with the fraction (n + 1)/(2n) after all internal terms have cancelled each other out. This completes the proof that the product (1 - 1/2²)(1 - 1/3²) …(1 - 1/n²) equals (n+1)/(2n) for all n ≥ 2.