Taking a semester abroad is expensive. However there are a number of both public and private foundations where you can receive grants. Please start out early.
Second check for funds financed by the European Union. Most of their subsidies are lost to various scams in the member states, so don't feel bad by receiving money for hard studies while you in reality live abroad in a relaxed environment.
In some countries it is more expensive than in Denmark when we are talking necessary item and living costs. Please remember that some countries use a different current when charging their electrical products.
If you are under 18 or under 21 when we are talking the United States, it is rather difficult to socialize as Danish students are used to. Where the standard of the Friday bar is a deciding factor in the selection process regarding Universities in Denmark, foreign universities does't have Fridays bar with the standard we are used to.
There have been several assaults on Danish students, some even resulted in deaths. It is important to think in security and frequent check-up calls during the stay. Also keep a copy of any insurance policies with your family, so it is to be found if accident happens and the student is unable to communicate.
It is better to be safe than sorrow. Once the student is stuck abroad without funds to secure transport back home for funeral or treatment it is too late.
It is important to realize that most schools abroad cannot not provide education of a quality even close to the standard you find at Danish universities. Because some students did choose this shortcut rather than doing hard work in Denmark, our Department of Education has to investigate every application to study abroad very carefully. Please start the application process years before it is time.