Exercise 356Calculus › Single-variable calculus › SequencesL1Added 2 Nov 2026

A telescoping product

#telescoping#products#factoring

Statement

Let $P_n=\displaystyle\prod_{k=2}^n\left(1-\frac1{k^2}\right)$ . Compute $P_n$ and its limit.

Hint

Factor $1-\frac1{k^2}=\frac{(k-1)(k+1)}{k^2}$ and write out the product.

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Factoring.
$P_n=\prod_{k=2}^n\frac{k-1}k\cdot\prod_{k=2}^n\frac{k+1}k.$

First product. $\frac12\cdot\frac23\cdots\frac{n-1}n=\frac1n$ .

Second product. $\frac32\cdot\frac43\cdots\frac{n+1}n=\frac{n+1}2$ .

Result.
$P_n=\frac{n+1}{2n}\longrightarrow\frac12.$