Exercise 1: Sum of differences between curves y = x^n as an infinite series

https://play.3speak.tv/embed?v=mes/ph3k5jrm

In this video, I show that the sum of differences between successive positive integer powers x^n for 0 ≤ x ≤ 1 can be written as an infinite series, and the resulting sum is equal to 1 or the unit square. This is the same series as in Example 6, thus this provides a geometric interpretation of the series 1/(n(n+1)).

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#math #calculus #series #desmos #education

Timestamps:

• Exercise 1: Geometric interpretation of Example 6 – 0:00
• Solution: Graphing curves of y = x^n with Desmos graphing calculator: https://www.desmos.com/calculator/ml00mwrvpi – 0:33
• Area between successive curves of x^n – 2:44
• Difference in areas as an integral – 6:30
• Difference is equal to the formula in the series from Example 6 – 9:03
• Sum of series of differences equals one or the unit square – 9:38

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