是write sb.a letter (to introduce oneself)还是(introducing oneself)?

题目

是write sb.a letter (to introduce oneself)还是(introducing oneself)?
我觉得应该是write sb.a letter to introduce oneself,因为是表示目的啊,但是答案却是write sb.a letter introducing oneself,请问为什么呢?
附题目原句：He wrote her a letter introducing himself and inviting her to correspond.（顺带问一下问啥后面的invite要用-ing?)

答案

write sb.a letter to introduce oneself,可以是表示目的,不定式作目的状语答案write sb.a letter introducing oneself,也是对的 现在分词 作定语 因为原句：He wrote her a letter introducing himself and invitin...