This proof is very similar to that of Rudin's Principles of mathematical analysis, and it is the same for some others of your videos. Are you taking these proofs from that book?
Also there's a condition that you haven't said in the video : the sequence a_n has to be, from a certain number N on, non-zero, otherwise it's not realy clear what is meant by the limit of the ratio, because some ratios doesn't exist (cause you are dividing by zero)
@@yichen8884 Yes, but think about what happens in the case a_n = 0 if 3 divides n, a_n = n if n divided by 0 has remainder 1 and a_n = n^2 if n divided by 0 has remainder 2 ( the sequence goes like 0,1,2^2,0,4,5^2,0,7,8^2, ... ). This sequence has infinitely many zeros and it doesn't even converge. What happens to |a_{n+1}| / | a_n | when n tends to infinity?
Haha, nice to see the uncut bug at 4:03. I'm happy to see that proof, I've wondered for quite some time how it could be proved.
Crystal clear! Thank you!
Cool, I hadn‘t seen this particular proof before. Do you have a video on the inequalities with ratios and roots used in the proof?
Yeah the video is called the pre ratio test
I am not really good at math, but it is really impressive and cool to watch anyway
It's amazing
Thanks for uploading this video.....really useful for my exam💯 thanks a lot
This proof is very similar to that of Rudin's Principles of mathematical analysis, and it is the same for some others of your videos. Are you taking these proofs from that book?
Also there's a condition that you haven't said in the video : the sequence a_n has to be, from a certain number N on, non-zero, otherwise it's not realy clear what is meant by the limit of the ratio, because some ratios doesn't exist (cause you are dividing by zero)
Took it from Ross
@@clementeromano5691 I wonder if, in the case of a_n becoming zeros after some N, the series converges automatically, as the sum is adding 0.
@@yichen8884 Yes, but think about what happens in the case a_n = 0 if 3 divides n, a_n = n if n divided by 0 has remainder 1 and a_n = n^2 if n divided by 0 has remainder 2 ( the sequence goes like 0,1,2^2,0,4,5^2,0,7,8^2, ... ). This sequence has infinitely many zeros and it doesn't even converge. What happens to |a_{n+1}| / | a_n | when n tends to infinity?
Ok. Thanks.
Hello sir