[size=150]The continued fraction of π (pi) is an infinite expression that represents the value of π as a series of fractions. A continued fraction is a unique way of expressing real numbers in terms of fractions with a specific pattern. The continued fraction of π is as follows:[br][br]π = [3; 7, 15, 1, 292, 1, 1, 1, ...][br][br]In this expression, the first term is 3, and then the subsequent terms form a repeating pattern. The sequence of terms {7, 15, 1, 292, 1, 1, 1, ...} repeats indefinitely.[br][br]To understand how this continued fraction works, let's see how it is formed step by step:[br][br]Step 1: The first term is 3.[br][br]π ≈ 3 + ...[br][br]Step 2: The next term is 7.[br][br]π ≈ 3 + 1/7 + ...[br][br]Step 3: The next term is 15.[br][br]π ≈ 3 + 1/(7 + 1/15) + ...[br][br]Step 4: The next term is 1.[br][br]π ≈ 3 + 1/(7 + 1/(15 + 1)) + ...[br][br]Step 5: The next term is 292.[br][br]π ≈ 3 + 1/(7 + 1/(15 + 1/(1 + 1/292))) + ...[br][br]This process continues indefinitely, with the sequence {7, 15, 1, 292, 1, 1, 1, ...} repeating in the continued fraction.[br][br]To approximate the value of π using the continued fraction, we can truncate the sequence after a certain number of terms. For example, let's use the first four terms:[br][br]π ≈ 3 + 1/(7 + 1/(15 + 1/1))[br][br]Now, we can perform the calculations:[br][br]π ≈ 3 + 1/(7 + 1/(15 + 1))[br]π ≈ 3 + 1/(7 + 1/16)[br]π ≈ 3 + 1/(7 + 0.0625)[br]π ≈ 3 + 1/(7.0625)[br]π ≈ 3 + 0.14084507[br]π ≈ 3.14084507[br][br]Using just the first four terms of the continued fraction, we obtain an approximation of π as 3.14084507. As we include more terms in the continued fraction, the approximation becomes more accurate. The continued fraction expansion of π is fascinating because it provides an infinite series of rational approximations for this famous irrational number.[/size]