I’m using the Twelve Days of Christmas (and perhaps a few extra days besides) to do something that I should have done a long time ago: collect past series of posts into a single, easy-to-reference post. The following posts formed my series on how I remind students about Taylor series. I often use this series in a class like Differential Equations, when Taylor series are needed but my class has simply forgotten about what a Taylor series is and why it’s important.
Part 1: Deriving the formulas for the th term of arithmetic and geometric sequences.
Part 2: Pedagogical thoughts on conceptual barriers that students often face when encountering sequences and series.
Part 3: The story of how young Carl Frederich Gauss, at age 10, figured out how to add the integers from 1 to 100 in his head.
Part 4: Deriving the formula for an arithmetic series.
Part 5: Deriving the formula for an arithmetic series, using mathematical induction. Also, extensions to other series.
Part 6: Deriving the formula for an arithmetic series, using telescoping series. Also, extensions to other series.
Part 7: Pedagogical thoughts on assessing students’ depth of understanding the formula for an arithmetic series.
Part 8: Deriving the formula for a finite geometric series.
Part 9: Infinite geometric series and Xeno’s paradox.
Part 10: Deriving the formula for an infinite geometric series.
Part 11: Applications of infinite geometric series in future mathematics courses.
Part 12: Other commonly-arising infinite series.