Finding ODE series solution coefficients

I am trying to solve an ODE by subbing in a series form and then looking individually at the coefficients of different powers of the variable. I’m looking at a general form of equation:

$$frac{mathrm{d}^2w}{mathrm{d}z^2} + f_1(z) frac{mathrm{d}w}{mathrm{d}z} + f_0(z)=0$$

I want to fill in the solution form $w(z) = e^{lambda z} z^mu sum_{s=0}^infty a_s z^{-s}$ and then collect coefficients of the various powers of $z$. However when I try to do this the series terms are not being multiplied with any $z$ terms which are multiplying the series from the outside. A simpler example of the problem I’m having is when I input say:

w[z_] = E^(λ z)z^μ  Sum[a[s] z^-s, {s, 0, Infinity}]  
Coefficient[z^5 w[z], z^μ]

The output of this is to give the coefficient as:
$$e^{lambda z} z^5sum_{s=0}^infty a_s z^{-s}$$

What I want is for the $z^5$ to multiply the $z^mu$ term and also term-by-term with the series terms, and then give me just the coefficient of the resulting $z^mu$ term. So the only corresponding term in the sum is the one of the form $e^{lambda z}z^5z^mu a_5 z^{-5}$, and therefore the coefficient I should be getting is just: $$e^{lambda z} a_{5}$$

Can I not do this type of manipulation or have I just done something wrong here?