Uploaded February 2022 | Updated September 2026, 1 week ago
We can group the terms in the divergent infinite sum 1-1+1-1+... to get a value of zero or a value of one, even though the series doesn't converge. This video gives an explanation of why grouping terms in Grandi's series doesn't give the correct answer.
Calculus Problems playlist: youtube.com/playlist?list=PLug5ZIRrShJGFne7YhMi-4eYsUKzkITao
0:00 Why the sum diverges
7:33 Why grouping fails
12:57 When grouping is valid
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Music: C418 - Pr Department
We can group the terms in the divergent infinite sum 1-1+1-1+... to get a value of zero or a value of one, even though the series doesn't converge. This video gives an explanation of why grouping terms in Grandi's series doesn't give the correct answer.
Calculus Problems playlist: youtube.com/playlist?list=PLug5ZIRrShJGFne7YhMi-4eYsUKzkITao
0:00 Why the sum diverges
7:33 Why grouping fails
12:57 When grouping is valid
Subscribe to see more new math videos!
Music: C418 - Pr Department


![Induction on Real Numbers
Video on supremum: https://youtu.be/o0TksrG5OsY
Paper on real induction: https://arxiv.org/pdf/1208.0973
Proof by induction is often taught as something that only works for integers, natural numbers, or whole numbers. However, theres an analogous concept to induction that applies to intervals on the real numbers, even though the real numbers are an uncountable set! This video explains and proves the result for the interval [0,1], which can be generalized to arbitrary closed intervals in the real numbers.
Topology playlist: https://www.youtube.com/playlist?list=PLug5ZIRrShJEGnUPxM1KUVOHkW4kN6QU_
0:00 The Theorem
4:36 Proof
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Music: C418 - Smooth Fall Induction on Real Numbers](https://i.ytimg.com/vi/bXTbGJxW_fE/mqdefault.jpg)







