she is writing a letter to a friend of hers,----him to atten

1.答案是：inviting.
2.
1）用不定式做目的状语未尝不可,但是此题关键在于句中那个逗号.
此时如果用不定式的话,因为逗号的缘故,则只能理解成结果状语（表出乎意料的结果）了.而做目的状语的不定式如果放在句子中间,则一般不用逗号隔开（当然不定式放句首时可以做目的状语）.
如：I'm writing to tell you something.
不定式做目的状语.
而：He rushed to the airport,to find his favorite star had left.
不定式做结果状语.
2）用inviting的确做伴随状语.
invite和write其实是同时（伴随）发生的,两个动作无先后关系,即“写信”的同时也“邀请”了.