Lec 38 | MIT 18.01 Single Variable Calculus, Fall 2007

The script begins with an introduction to MIT OpenCourseWare, highlighting its mission to provide free, high-quality educational resources. The speaker invites viewers to support and explore additional materials on the platform. Transitioning to a classroom scenario, the professor presents a physical challenge involving stacking blocks in a way that defies balance, aiming to illustrate concepts related to series in mathematics. The challenge is to build a stack with blocks extending further left without falling, prompting a classroom vote on the feasibility of the task with a given number of blocks.

The professor delves into the concept of convergence and divergence in the context of the block stacking challenge. He explains that the outcome of the challenge could reveal an interesting mathematical property: either there is a limit to how far the blocks can extend (indicating convergence), or there isn't a limit (suggesting divergence). The professor then demonstrates a strategy to stack the blocks, emphasizing the importance of starting from the top down, contrary to the usual bottom-up approach. This method, known as the greedy algorithm, aims to optimize at each step, leading to a successful but precarious stack of blocks.

The script continues with a detailed mathematical analysis of the block stacking strategy. The professor introduces a thought experiment to calculate the center of mass for a stack of blocks, using it to determine the optimal placement of each subsequent block. He establishes a recurrence formula to calculate the center of mass for an increasing number of blocks, revealing a pattern that relates to the harmonic series. The professor then engages the audience in a discussion about the potential extent of the block stack, using the harmonic series to estimate the number of blocks needed to reach a certain distance.

The professor discusses the implications of infinity in the context of the block stacking problem, highlighting the logarithmic growth rate of the harmonic series. He points out that while the series diverges, indicating that theoretically, one could stack blocks infinitely far, the practical application requires an enormous number of blocks to achieve even a modest distance. The professor uses the example of stacking blocks across a room, calculating that it would require a number of blocks whose height would be twice the distance to the moon, emphasizing the slow growth of logarithmic functions.

The script transitions to a new topic: power series. The professor introduces the concept by revisiting the geometric series, which sums to 1/(1-x) for values of x less than 1. He provides a proof for this formula, demonstrating how multiplying the series by x and subtracting the original series from the result leads to the formula. The professor acknowledges that the proof is flawed when x equals 1, as it leads to an undefined state, but is valid for convergent series where x is less than 1.

The professor explains the importance of convergence in power series, emphasizing that it allows for mathematical manipulations that would otherwise be invalid. He introduces the concept of the radius of convergence, a value that determines the interval within which a power series converges. The professor clarifies that while there are methods to calculate the radius of convergence, they will not be discussed in detail, as it will be evident for any given series. He notes that the series will converge exponentially fast within the radius of convergence and diverge when x is greater than the radius.

The script includes an interactive segment where the professor addresses audience questions about the nature of the convergence and how to identify it. He also explains why the series converge exponentially fast, attributing it to the power nature of the terms involved. Another question prompts the professor to discuss the method of finding the radius of convergence, which he states will be evident in any given series without needing complex calculations. The professor then outlines the importance of power series in representing a wide range of functions, akin to how decimal expansions represent real numbers.

The professor discusses the operations that can be performed with power series, drawing parallels with polynomials. He mentions addition, multiplication, substitution, and division as operations that are analogous for both. The focus then shifts to differentiation and integration, which are particularly relevant for calculus. The professor outlines the rules for differentiating and integrating power series, providing formulas for these operations. He also introduces Taylor's formula, which is a method to express functions in terms of their derivatives evaluated at a point, typically zero, multiplied by powers of x and divided by factorials.

The script concludes with examples of Taylor series for the exponential function e^x, and the trigonometric functions sin(x) and cos(x). For e^x, the professor shows that all derivatives evaluated at zero are one, leading to a series of 1/n! times x^n. For sin(x) and cos(x), the series are expressed with alternating signs and powers of x, with sin(x) involving only odd powers and cos(x) involving even powers. The professor emphasizes the similarity of these formulas, suggesting that recognizing their patterns will aid in memorization. The session ends with a note on practicing differentiation with these series in future lessons.