I would like to add that singles are not always sitting at home waiting to be invited. Sometimes we actually (gasp!) make our own meals.
When you invite a single, and she respectfully declines, change the topic. Pushing will only push her away. At least it will if I’m the one you’re pushing.
And remember, until your friends, who are your age, are having grandchildren, don’t call yourself an “older” single. Thanks.