Does altitude influence the flight range of an engine driven aircraft? I'm reading a chapter about flight performance during the cruise phase and the influence of altitude. 

My book (Aerodynamica, prestatieleer en vliegtuigtechniek by Bas Vrijhof on page 155, written in Dutch) describes the following: 


  Naarmate de hoogte toeneemt, neemt de luchtdichheid af. De afnemende
  luchtdichtheid heeft een effect op het benodigde vermogen, en dus op
  de prestaties. Om bij een lagere luchtdichtheid te vliegen is meer
  vermogen nodig. Om bij een lagere luchtdichtheid voldoende lift te
  houden moet de TAS toenemen, dat kost vermogen.


Translated to English: 


  Airdensity decreases with altitude. A decrement in airdensity influences flight performance. To fly at a lower airdensity more engine power is needed. To maintain enough lift at lower airdensity the TAS needs to increase, this comes with the cost of engine power.  


This is totally clear to me. Then the book says. 


  Endurance...
  Meer vermogen betekent een hogere brandstofverbruik. De endurance neemt daarom af.


Translated: 


  Endurance (amount of fuel per hour), more engine power means a higher fuel consumption. The endurance decreases with an increase in engine power.


Again, this is also clear to me. 


  Range...
  De range verandert niet bij toenemende hoogte. De range is niet afhankelijk van de vlieghoogte. Wel neemt de Vmax range toe als de hoogte toeneemt.


Translated: 


  Range (maximum flight distance, amount of fuel per km) doesn't change with an increase in altitude. However, the Vmax range increases with an increase in altitude.


The last sentence is confusing me. 
Because my instincts say that a higher fuel consumption comes with a shorter flight distance. So my question is: 

Is it really true that the altitude doesn't influence the flight range? Does the book has an error?
Your book may not be wrong, but it is misleading, because the answers depend on the the aircraft type, and the type of engine. (piston, turboprop, turbojet, turbofan)

In general: 

All aircraft have the best range at higher altitudes.

Piston powered aircraft have the best endurance at low atitude.

Jet powered aircraft have the best endurance at their optimum altitude. (not too low and not too high)