Lec 37 | MIT 18.01 Single Variable Calculus, Fall 2007

The professor begins by discussing improper integrals of the second kind, which have a singularity at a finite point. The concept is introduced with the integral from 0 to 1 of f(x), which is defined as the limit of the integral from a to 1 of f(x) dx as a approaches 0 from above. The idea is illustrated with a graphical representation of a function approaching the singularity at point 1. The professor explains the convergence of such integrals, stating that they converge if the limit is finite and diverge otherwise, with examples including the integral of 1 over the square root of x which converges, and the integral of 1 over x which diverges.

The lecture continues with a deeper exploration of improper integrals, contrasting those with singularities at finite points versus infinity. The professor emphasizes the importance of understanding the behavior of functions as x approaches 0 and infinity, highlighting the differences with examples such as 1/x^(1/2) which is smaller than 1/x and smaller than 1/x^2. The professor illustrates the concept with a graphical representation, showing how the area under the curve can be finite or infinite depending on the function's behavior, and how integrals associated with functions that tend towards infinity or 0 can be divergent.

The professor provides a detailed analysis of improper integrals using specific examples, such as the integral of 1/(x-3)^2 from 0 to infinity. The integral is split into two parts to understand its behavior, with the first part being infinite due to the square in the denominator as x approaches 3, and the second part being finite. The conclusion is that while the individual parts may have different properties, the overall integral diverges if it contains an infinite part. The professor also addresses questions from the audience regarding the divergence of improper integrals and the importance of recognizing singularities.

Moving away from improper integrals, the professor transitions to the topic of infinite series, emphasizing their importance in dealing with infinity. The most important and useful series discussed is the geometric series, introduced through a concrete example. The professor illustrates the convergence of a geometric series to a finite sum using a number line, highlighting the paradox of Zeno's race, which questions the possibility of reaching a point when the number of steps is infinite.

The professor delves into the general case of geometric series, emphasizing the formula 1/(1-a) for their sum, which only applies when the absolute value of a is less than 1. The lecture contrasts this with cases where a equals 1 or -1, leading to divergence. The professor also touches on the historical and theoretical implications of these series in number theory, where divergent series can sometimes be given meaning under different mathematical frameworks.

The lecture continues with a systematic approach to understanding series convergence and divergence. The professor introduces the notation for partial sums, S_N, and the full limit, S, and explains how the existence of the limit indicates convergence, while its non-existence indicates divergence. The professor also discusses the analogy between series and integrals, using the example of the sum of 1/n^2 from 1 to infinity, which is convergent and can be evaluated, in contrast to the more complex sum of 1/n^3, which is also convergent but more challenging to evaluate.

The professor illustrates the connection between series and integrals using the example of the sum of 1/n from 1 to infinity, which corresponds to the integral from 1 to infinity of dx/x. The lecture demonstrates that the Riemann sum, with delta x equal to 1, can be used to show that the series diverges by comparing it to the integral. The professor also explains the concept of integral comparison and limit comparison tests for determining the convergence or divergence of series.

The professor discusses the application of comparison tests to series, providing examples to illustrate how these tests can be used to determine the convergence or divergence of various series. The lecture emphasizes the importance of recognizing the main term in a series and using it to compare with known convergent or divergent series. The professor also teases a demonstration for the next lecture, involving stacking blocks in a specific arrangement, to be explored further.