Approach 1: (no longer works)
https://plus.google.com/s2/photos/profile/<user_id>?sz=<your_desired_size>
Approach 2: (each request counts in your api rate limits which is 10k requests per day for free)
https://www.googleapis.com/plus/v1/people/<user_id>?fields=image&key={YOUR_API_KEY}
with the following response format:
{ "image": { "url": "lh5.googleusercontent.com/-keLR5zGxWOg/.../photo.jpg?sz=50"; } }
Approach 3: (donot require api key)
http://picasaweb.google.com/data/entry/api/user/<user_id>?alt=json
in the json response you get a property named "gphoto$thumbnail", which contains the profile picture url like the following:
http://lh6.ggpht.com/-btLsReiDeF0/AAAAAAAAAAI/AAAAAAAAAAA/GXBpycNk984/s64-c/filename.jpg
You may notice in the url the portion "s64-c" which means the image size to be 64, I've tried using other values like "s100-c" and they worked. Also if you remove the "s64-c" part and append the "?sz=100" parameter, that will also work as of now. Though this is not very good way of getting the profile picture of a gplus user, but the advantage is it do not require any api key.