以下兩句都是對的，用分詞，表伴隨；用不定式，表目的：

My mother wrote giving me details about the party.　　

My mother wrote to give me details about the party.

對于目的意味比較強的句子，宜用不定式。如：

He never went again, nor did he write to apologize. 他再也沒去，也沒去信致歉。

陳老師回答的超贊，一下子解決了好幾個問題。