Proof: Sequence n^2 Diverges to Infinity | Real Analysis

[@WrathofMath]

Merry Christmas! I hope you all have a splendid and jolly day today, and I hope you get some cool math books - I know I will! 🎅

[@ameliaburnell6563]

This made it so much easier and more understandable, thank you!

[@WrathofMath]

Glad to hear it, thanks for watching!

[@EpicMathTime]

4:30 Hah, I was going to comment on this before you got to it. It doesn't really serve any purpose to have N as a natural number, but it seems to be pretty regularly done. Maybe it is called a natural number when the definition is sometimes presented. I remember grading an "advanced calculus" (what my university called undergraduate real analysis) and one of the students would do things like "let N = ceiling[sqrt(M)]". I appreciated the attention to detail. I was just the paper grader, so I don't actually know how they were taught in the class. Maybe "N" just screams natural number to people.

[@WrathofMath]

Indeed, I like to neglect the N being a natural number part of the definition since it is needless and less clunky to allow it to be any real number. But at the same time, I like to invoke its spirit so we can refer to the "Nth term of the sequence" and have that actually mean something, just because I think the best way to explain what we're doing is "showing something holds for every term after the Nth term", or "for the Nth term and onwards" depending on whether we're using greater than or /geq. I'll probably say N is natural most of the time just so I can explain things that way and hopefully prevent confusion haha. And N definitely screams natural number to me. I remember the silly attention to detail I and my peers had in college - the example that makes me cringe is showing something is even, showing it equals say 2(j^2+j) and then saying let (j^2+j)=k and notice k is an integer, and so 2(j^2+j) = 2k is even. 😂

[@Dev_skoll]

This channel has a future. I see it. Like the feeling of Three blue one brown and Smarter Everyday.

[@WrathofMath]

Thank you for watching and I appreciate your optimism, Bryce! Hopefully 2021 brings good things! I know I'll do my best to bring more great math content!

[@monkklein8282]

I just discovered you amazing channel today! Happy with my Christmas gift!

[@WrathofMath]

So glad you found the channel and like what you see! Thanks for your support and let me know if you ever have any video requests!

[@adoxographer]

Should the inequality be strict for N > sqrt(M), as sqrt(M) may not be a natural number?

[@mahmoudalbahar1641]

Many thanks for this good video.

[@WrathofMath]

My pleasure, thanks as always for your support!

[@TranquilSeaOfMath]

Interesting that almost a year after you made this video, I was thinking of this and found your video. I came to this problem as I was originally thinking of the series n^2. I do like getting mathematics books for Christmas too. Merry Christmas!

[@WrathofMath]

That is funny, glad you found it! I think I'll put "sequence" in the title, not sure why I didn't - I usually do if it is relevant but sometimes it makes the titles too long. Merry Christmas to you as well! Since last year I have moved, but I still put my Christmas set together for math lessons kzbin.info/www/bejne/j3eofK2AorF1ptE

[@TranquilSeaOfMath]

@@WrathofMath I understand not wanting titles to be too long. We do have the description box to put in a... good description. 😀

[@WrathofMath]

Haha precisely! It's always a balance between title, thumbnail, and description! Thumbnail is best really since it's the only place you can go wild with math notation.

[@reject_green494]

Such a great video! I have a question though what would happen if it was -n^2 instead of n^2?

[@monkeytrollhunter]

I have an exam in 2 hours and I'm watching this!!! AHHHHHHHHHHHH!!!!

[@WrathofMath]

DO YOUR BEST!!!

[@monkeytrollhunter]

@@WrathofMath I think I got my ass kicked, but I will study harder next time. I will be back for more.

[@WrathofMath]

That's the right vengeful spirit to have! Ass kickings happen to the best of us, especially with challenging mathematics!

[@Bobby-nq4qs]

But if you do the same argument on a_n= 1/n, would it give you the same result? Since 1/n > M?

[@WrathofMath]

Thanks for watching and for the question! No, if M is anything greater than 1, then we have 1 < M. Since n is a positive integer, dividing by n only makes things smaller, so then clearly 1/n < M. Does that help?

[@mlchior]

How exactly do you proceed if you have given n^2 - n instead of n^2? It would be great if you could help me :D

[@WrathofMath]

Thanks for watching and the question! A couple possibilities... it could easily be shown that if a_n and b_n both diverge to positive infinity, then a_n * b_n does as well. Then apply that to n*(n-1). Alternatively, just use a direct proof on n*(n-1). Take an M>1, if n>M then certainly n*(n-1) > M, and we're done.

[@mlchior]

@@WrathofMath Thank you 👍

[@westxlcr]

I appreciate it!

[@WrathofMath]

I appreciate you!