Uploaded February 2026 | Updated September 2026, 2 weeks ago
In this video, we investigate the Faulhaber formulas for the sum of the first n positive powers of p (where p is an integer). The sum of integers, squares, and cubes, all have several beautiful visual proofs. Here we include one each of the most classic.
For the higher powers, it can be difficult to visualize the sums. However, we can uncover these formulas visually if we inductively use the powerful of summation by parts, which does have a visual argument. We show how to do this for the sum of the fourth powers and suggest how to extend to the sum of the fifth powers. Can you create more formulas? Can you use summation by parts to produce other interesting formulas?
If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks!
The diagrams and animations in this video are based on several well-known visual proofs. Sources are below.
The sum of the first n positive integers was known to the ancient greeks according to Roger Nelsen's first proof without words book:
bookstore.ams.org/clrm-1
To see more about summing positive integers, check out
youtu.be/eHbtc50-qXo
The sum of squares proof is based on a visual proof by Man-Keung Siu from the March 1984 issue of Mathematics Magazine (jstor.org/stable/2689589), page 92.
For standalone versions see:
youtu.be/WidzHiUFWNA
or
youtube.com/shorts/PVH5uxHFIGQ
The sum of cubes proof is based on independently discovered, separate visual proofs by J. Barry Love from the March 1977 issue of Mathematics Magazine (jstor.org/stable/2689727), page 74, and Alan L. Fry from the January 1985 issue of Mathematics Magazine (jstor.org/stable/2690228) , page 11.
For a standalone version of this animation see:
youtu.be/YQLicI8R4Gs
or
youtube.com/shorts/O_5CnZh-G7Q
Finally, the visual proof for summation by parts is based on a diagram from Roger Nelsen's book Cameos for Calculus :
bookstore.ams.org/clrm-49
#math #manim #mathvideo #sumofintegers #sumofsquares #sumofcubes #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #finitesums #discretemath #faulhaber #faulhaberformulas #sumsofpowers
To learn more about animating with manim, check out:
https://manim.community
In this video, we investigate the Faulhaber formulas for the sum of the first n positive powers of p (where p is an integer). The sum of integers, squares, and cubes, all have several beautiful visual proofs. Here we include one each of the most classic.
For the higher powers, it can be difficult to visualize the sums. However, we can uncover these formulas visually if we inductively use the powerful of summation by parts, which does have a visual argument. We show how to do this for the sum of the fourth powers and suggest how to extend to the sum of the fifth powers. Can you create more formulas? Can you use summation by parts to produce other interesting formulas?
If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks!
The diagrams and animations in this video are based on several well-known visual proofs. Sources are below.
The sum of the first n positive integers was known to the ancient greeks according to Roger Nelsen's first proof without words book:
bookstore.ams.org/clrm-1
To see more about summing positive integers, check out
youtu.be/eHbtc50-qXo
The sum of squares proof is based on a visual proof by Man-Keung Siu from the March 1984 issue of Mathematics Magazine (jstor.org/stable/2689589), page 92.
For standalone versions see:
youtu.be/WidzHiUFWNA
or
youtube.com/shorts/PVH5uxHFIGQ
The sum of cubes proof is based on independently discovered, separate visual proofs by J. Barry Love from the March 1977 issue of Mathematics Magazine (jstor.org/stable/2689727), page 74, and Alan L. Fry from the January 1985 issue of Mathematics Magazine (jstor.org/stable/2690228) , page 11.
For a standalone version of this animation see:
youtu.be/YQLicI8R4Gs
or
youtube.com/shorts/O_5CnZh-G7Q
Finally, the visual proof for summation by parts is based on a diagram from Roger Nelsen's book Cameos for Calculus :
bookstore.ams.org/clrm-49
#math #manim #mathvideo #sumofintegers #sumofsquares #sumofcubes #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #finitesums #discretemath #faulhaber #faulhaberformulas #sumsofpowers
To learn more about animating with manim, check out:
https://manim.community
