A daughter-in-law rendering services to her father-in-law as she does to her own father
Can a daughter-in-law render services to her father-in-law as she does to her own father?
The relationship between a man and his daughter-in-law prohibits marriage between them for life, regardless of whether her marriage to his son continues or ends by divorce or by her husband's death. Therefore, he can serve as her mahram (a relative whom she is not permitted to marry) on any journey including pilgrimage and umrah.
If, by ‘rendering services’ the inquirer means looking after her father-in-law, particularly when he is ill, then surely she can and should do so. The same is the case if it means doing whatever that makes him comfortable at home. While this is not a duty upon her, it is an act of kindness which every daughter-in-law should be encouraged to do.
Having said that, it should be added that the Prophet (peace and blessings be upon him) warned against relations between in-laws getting too familiar. This means that Islamic standards of propriety should always be maintained between in-laws so as not to yield to temptation and do anything prohibited.
And God Almighty knows best.