The function of pumps in forming the resting potential I am confused by the following. Apparently the resting potential of -65mV is reached when the two forces, diffusion and electrical gradient are in equilibrium. So why does the book say 

"The electrical potential difference across the membrane [..] charge is maintained by the work of the ion pumps."

Why should it be "maintained by the pumps" if it is an equilibrium anyway?

One possible answer I can imagine is: the equilibrium is too unstable and so the pumps are stabilizing it. But I have no idea if this is correct. Thanks.
Apparently the resting potential of -65mV is reached when the two forces, diffusion and electrical gradient are in equilibrium.


That is not true. You are confusing a neuron's resting potential with what is actually a type of ion's equilibrium potential. 

An equilibrium potential for a particular type of ion--such as Na+ or K+--is the exact voltage across the membrane that exactly opposes the diffusional force for that ion, and thereby prevents a net flow of that ion across the membrane. So, for example, in the case of Na+ (sodium ion), in mammalian neurons, a typical value given in textbooks is something like +55 mV. In other words, the interior of the neuron needs to be 55 mV (relative to the outside of the neuron) in order to prevent all the Na+ ions from rushing into the neuron. There are equilibrium potentials for all the ions that are used in neurons to influence the voltage across the membrane: Na+, K+, Cl-, Ca++. If a fictional neuron were to be at the equilibrium potential for Na+, and there were no other ions at play, there would be no need for pumps to maintain any concentration gradient of Na+., because, by definition, there would be no net flow of charge, and so the situation would be stable.

However, the resting potential refers to the voltage of the neuron when it is at rest--not having an action potential--but that voltage is going to be some "weighted average" of the equilibrium potentials of all of the types of ions: Na+, K+, Cl-, and Ca++. Of these, K+ (potassium ion) dominates, and so we find that typical values for the resting potential of neurons is around -65mV, which is closer to the equilibrium potential for K+ (-90mV), than for Na+ (+55mV) or Ca++ (+155mV!). 

But since the typical resting potential of the neuron actually is not equal to any of the equilibrium potentials for any type of ion, that means by definition all of the ions are going to have net movement across the membrane. If ion channels are left open and ions are therefore allowed to flow across the membrane, Na+ is going to flow in, K+ is going to flow out, Ca++ is going to flow in, etc. (I'm leaving Cl- out for simplicity).

Since these ions are flowing with a net direction (either into or out of the neuron), that means concentrations are changing; e.g., K+'s large concentration gradient is getting smaller as it flows out of the neuron. Over time, these ion flows move the concentrations of ions towards equality on both sides of the membrane, and therefore move the neuron's resting membrane potential toward 0 mV. In some sense, the "battery is running down". This would render the neuron incapable of having an action potential, and thus acting as a signaling machine, and therefore must be prevented.

In order to prevent that rundown of concentration gradients, ion pumps actively maintain the concentration gradients of the ions. One of the relevant ones here is the Na+/K+ exchanger, which uses ATP to power its action of pumping in two Na+ ions for every three K+ ions it pumps out. Using ATP requires producing ATP, which requires glucose, and so just maintaining "ready to go neurons" is metabolically demanding.