7の平方根
Each convergent is a best rational approximation of 7 {\displaystyle {\sqrt {7}}} ; in other words, it is closer to 7 {\displaystyle {\sqrt {7}}} than any rational with a smaller denominator. Approximate decimal equivalents improve linearly (number of digits proportional to convergent number) at a rate of less than one digit per step:
2 1 = 2.0 , 3 1 = 3.0 , 5 2 = 2.5 , 8 3 = 2.66 ... , 37 14 = 2.6429... , 45 17 = 2.64705... , 82 31 = 2.64516... , 127 48 = 2.645833... , ... {\displaystyle {\frac {2}{1}}=2.0,\quad {\frac {3}{1}}=3.0,\quad {\frac {5}{2}}=2.5,\quad {\frac {8}{3}}=2.66\dots ,\quad {\frac {37}{14}}=2.6429...,\quad {\frac {45}{17}}=2.