Should Powerful Countries Concern Themselves with the Well-Being of Less Fortunate States?

I hold the belief that all people, and all states, are self-interested. A state’s well-being is of the highest priority, which is logical considering the people in leadership positions are elected by the populace of that specific nation to serve their needs. But the question is whether or not powerful states should look after the needs of weaker nations. I believe that if the powerful state stands to benefit or prevent dangerous situations by providing aid, it should assist that weaker nation out of its own self-interest.
The greatest problem in the eyes of powerful nations regarding weak states is the threat to stability these underdeveloped nations pose. In most cases, poor countries have unstable governments, largely due to the fact discontent runs high in nations that can’t provide basic services for their citizens. Unstable nations tend to have a dangerous mix of factors: a large part of the populace that is very young (due to high birth rates and poor health care); there is little formal education; and weapons are often widely available. This potent concoction can spill into neighboring nations and can be felt around the world. Al-Qaeda is a prime example of a neglected group of people who turned on the developed world to spread a radical ideology and vent their anger. Saudi construction heir and Egyptian doctor aside, the majority of Al-Qaeda’s organization are from poor, disenfranchised parts of the world such as Afghanistan, Chechnya, and Yemen. Individuals brought up knowing they were poor and helpless to stop it look for ways to get revenge against those who they feel are responsible for their suffering. This combines with radical religious beliefs and a readily available supply of firearms, and soon wealthy nations find their national security at risk by terrorists.
So, should powerful states support weaker ones? I personally believe basic aid concentrated in education, infrastructure, and health care is the key. Instability is inevitable in countries where these three things are not widely available to the public. Education and health care are long-term services that help calm discontent among people and infrastructure raises the quality of living and connects communities, allowing for expanded local business. Wealthy nations should assist poor nations in these fields to increase stability, reducing the chances of discontent around the world spilling into their nations in the form of a terrorist attack or other act of aggression. Military aid to nations with authoritarian regimes breeds ill will among the populace, and encourages people to turn personal vendettas into global acts of terror. I believe the United States and other leaders of the free world should avoid giving military aid to nations with track records of human rights abuses to distance itself from the image of being supportive of pro-western dictators.
So yes, I do believe that wealthy nations should assist weaker states. However, I am in favor of doing it when it is deemed in the best interest of the powerful country. There is no need to break the bank trying to iron out the finer points of humanitarian aid to other nations, but powerful nations should concentrate assistance in specific areas to maintain relative stability in the third world.