Beyond Mere Convergence

In this article, I suggest that calculus instruction should include a wider variety of examples of convergent and divergent series than is usually demonstrated. In particular, a number of convergent series, such as ∑ k≥1 k3 2k , are considered, and their exact values are found in a straightforward manner. We explore and utilize a number of mathematical topics, including manipulation of certain power series and recurrences. During my most recent spring break, I read William Dunham’s book Euler: The Master of Us All [3]. I was thoroughly intrigued by the material presented and am certainly glad I selected it as part of the week’s reading. Of special interest were Dunham’s comments on series manipulations and the power series identities developed by Euler and his contemporaries, for I had just completed teaching convergence and divergence of infinite series in my calculus class. In particular, Dunham [3, p. 47-48] presents Euler’s proof of the Basel Problem, a challenge from Jakob Bernoulli to determine the