Doubling the length of a solenoid doubles its inductance. Two identical solenoids in series have up to four times the inductance due to M. Why? The inductance of a long solenoid with $N_o$ turns and a length $l_o$ is
$$L_o=\pi r^2 \mu_0\frac{N_o^2}{l_o}$$
If I now make a new solenoid, $L_{new}$, with double the length of the original ($l_{new}=2l_o$) and double the number of turns as well (in order to keep the turns per unit length constant), I clearly have a new inductor with double the inductance of the original solenoid so that $L_{new}=2L_o$. But doubling the length with constant $\frac{N}{l}$ is the same as putting two long solenoids of inductance $L_o$ together in series (with perfect coupling). Yet I am told that when two inductors ($L_1$ and $L_2$) are placed in a series aiding configuration, the combined inductance is (Fundamentals of electric circuits, by Alexander):
$$L_{Tot}=L_1+L_2+2M$$
where $M$ is the mutual inductance between the two inductors with $M=k\sqrt{L_1\cdot L_2}$ ($k$ being the coupling coefficient). Suppose these inductors both have inductance $L_o$. Then when placed in a series aiding fashion, we have
$$L_{Tot}=2L_o+2\cdot k\cdot L_o$$
The further these inductors are apart (still being in series though), presumably the smaller the coupling coefficient $k$ so that in the limit that they are infinitely far apart, we have $k=0$ and $L_{Tot}=2L_o$. But conversely, the closer they are the larger $k$ gets, so that when the end of the first just touches the beginning of the second, we surely have $0 <k\leq 1$. Thus, in this case we must have that $L_{Tot}>2L_o$. But this case is identical to the first case where we simply doubled the length of the original inductor and doubled the number of turns to get $L_{new}=2L_o$ so we have a contradiction. How can this be? How can we get two different values for the inductance of the same solenoid? A visual representation of my issue is given below:

 A: 
But this case is identical to the first case where we simply doubled the length of the original inductor and doubled the number of turns to get $L_{new}=2L_0$ so we have a contradiction. How can this be? How can we get two different values for the inductance of the same solenoid?

The two situations are not identical:

*

*in case of a single solenoid of length $2l_0$ there is the same current in every cross-section of the wire forming the solenoid (if the current is constant or the solenoid is sufficiently short to neglect the spatial variation of the current)

*in case of two solenoids, each of them affects the current in the other one, thus changing the magnetic field in the other solenoid and affecting the overall inductance. This is what is behind the mutual inductance term in the equations in the OP. The answer here (i.e., the value of $k$) would depend on how exactly the two solenoids are connected in the circuit.

A: The formula you're using is an approximation that assumes $l>>r$. But in that approximation, there is negligible mutual inductance, since the flux spreads out at the end on a scale of $\approx r$.
