<P> If b is an integer base and k is an integer, </P> <Dl> <Dd> 1 k = 1 b + (b − k) 1 b 2 + (b − k) 2 b 3 + (b − k) 3 b 4 + (b − k) 4 b 5 + ⋯ + (b − k) N − 1 b N + ⋯ (\ displaystyle (\ begin (aligned) (\ frac (1) (k)) \ & = (\ frac (1) (b)) + (\ frac ((b-k) ^ (1)) (b ^ (2))) + (\ frac ((b-k) ^ (2)) (b ^ (3))) + (\ frac ((b-k) ^ (3)) (b ^ (4))) + (\ frac ((b-k) ^ (4)) (b ^ (5))) + \ cdots + (\ frac ((b-k) ^ (N - 1)) (b ^ (N))) + \ cdots \ \ (6pt) \ \ \ end (aligned))) </Dd> </Dl> <Dd> 1 k = 1 b + (b − k) 1 b 2 + (b − k) 2 b 3 + (b − k) 3 b 4 + (b − k) 4 b 5 + ⋯ + (b − k) N − 1 b N + ⋯ (\ displaystyle (\ begin (aligned) (\ frac (1) (k)) \ & = (\ frac (1) (b)) + (\ frac ((b-k) ^ (1)) (b ^ (2))) + (\ frac ((b-k) ^ (2)) (b ^ (3))) + (\ frac ((b-k) ^ (3)) (b ^ (4))) + (\ frac ((b-k) ^ (4)) (b ^ (5))) + \ cdots + (\ frac ((b-k) ^ (N - 1)) (b ^ (N))) + \ cdots \ \ (6pt) \ \ \ end (aligned))) </Dd> <P> For example 1 / 7 in duodecimal: </P>

Write the number in decimal recurring form 13/3