For what value of k do the following equations have an infinite number of solutions: kx + 3y = 2k+1, 2(k+1)x + 9y = 7k+1?

From equation (1): $a_1 = k$, $b_1 = 3$, $c_1 = 2k+1$
From equation (2): $a_2 = 2(k+1)$, $b_2 = 9$, $c_2 = 7k+1$

Simplify the right side:

Cross-multiply:

Subtract $2k$ from both sides:

Substitute $k=2$ into the ratio $\frac{3}{9} = \frac{2k+1}{7k+1}$:
Left side:

Right side:

Since $\frac{1}{3} = \frac{1}{3}$, the value $k=2$ satisfies all conditions for the system to have an infinite number of solutions.

The value of $k$ is $\boxed{\text{2}}$.