They die as their organs gradually shut down. The body forgets how to work, so to speak. I believe they are generally considered to have died of Alzheimer's unless some other condition takes them first, such as chronic COPD or other heart disease or stroke. However, a doctor would have to give you a more definite answer.
Most individuals live an average of eight to ten years after diagnosis. According to the Center for Disease Control, Alzheimer’s disease is the seventh leading cause of death in the United States. Once an individual reaches the latest stage of Alzheimer’s disease the damage to the brain is pervasive. A person in this stage is totally dependent on others and most likely cannot perform most tasks without assistance. When an individual has such limitations of functioning and impaired bodily functions, other illnesses can affect them to a greater degree. For example, pneumonia is a frequent complication of dementia and can be the immediate cause of death.
People with Alzheimer's die as a result of the following:
*From another medical condition such as heart disease or cancer.
*from the interplay of another illness and Alzheimer's where their dementia has not affected their functioning to a great degree.
*from end stage Alzheimer's where the disease impacts on all areas of their life and they die from the complications of the condition. Pneumonia is one such end condition.
I wrote a sharepost about dying from Alzheimer's Disease which gives more information and which you might find helpful
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