i though of neat solution to the lim showed in this video: https://www.youtube.com/watch?v=89d5f8WUf1Y my solution:what you think?

This is de alamet's test for sequence

Radius of convergence concept

Interesting. Can you show the part: If limit of a_{n+1}/a_{n} = L, then the n-th root of a_n also goes to L?