Biophysics Problem -- Summation Issues

[nisler.1]

Homework Statement

The average number of mRNAs in the cell at any time t is <m>(t) = Σ m * p(t). Sum over all the differential equations derived in a) in order to obtain a differential equation for <m>(t)

Homework Equations

So the differential equation I got in a) was dp/dt = (-k_p * P_m) - (m * k_m * P_m) + (k_p * P_m-1) + ((m+1) * k_m * P_m+1)

That would make d<m>/dt = Σ m * dP_m/dt = Σ m * ((-k_p * P_m) - (m * k_m * P_m) + (k_p * P_m-1) + ((m+1) * k_m * P_m+1))

What I need is d<m>/dt without any p(t)'s left on the right hand side

The Attempt at a Solution

I first separated the k_p and k_m terms:

d<m>/dt = Σ m * k_p * (-P_m + P_m-1) + Σ m * k_m * ((-m * P_m) + ((m+1) * P_m+1))

Now I know I have to manipulate this using substitutions such as m' = m -1, m = (m-1) + 1, and shifting of indices, but I am completely unsure how to proceed. Any help would be greatly appreciated.
Sorry I forgot to mention, the summations are from m = 0 to infinity

 

[mfb]

It often helps to write down the first few terms of those sums explicitely. You'll see what cancels and what does not.