Improvidently is how you do something when you do not think about later — you spend or decide without saving or planning. He improvidently gave away the last of the money. Same meaning as improvident, only this form tells you about the action, not the person. Same formal, old-fashioned flavour. Students almost never need this word in speech; if you know the adjective, you can build this one by adding -ly. In ordinary talk, 'without thinking ahead' or 'without saving' is clearer.