Smallest maximum rounding error

High school · ●●○○○ 20 pts

The sum of the following seven numbers is exactly 19:

a_1 = 2.56,

a_2 = 2.61,

a_3 = 2.65,

a_4 = 2.71,

a_5 = 2.79,

a_6 = 2.82,

a_7 = 2.86.

Each

a_i

is approximated by an integer

A_i,

for

1 \le i \le 7,

such that the sum of the

A_i

's is also

19.

Let

M

be the maximum of the seven errors

|A_i - a_i|.

What is the smallest possible value of

M

? Give your answer as a decimal.