Tests for Convergence, Pt. 1 The comparison test, the limit comparison test, and the integral test.

Comparing series. . . Consider two series , with for all k. In this presentation, we will base all our series at 1, but similar results apply if they start at 0 or elsewhere.

Comparing series. . . Note that: What does this tell us? Consider two series , with for all k. How are these related in terms of convergence or divergence? Note that: What does this tell us?

Comparing series. . . Note that: Consider two series , with for all k. Note that: Where does the fact that the terms are non-negative come in? What does this tell us?

Series with positive terms. . . Since for all positive integers k, So the sequence of partial sums is . . . Non-decreasing Bounded above Geometric

Suppose that the series converges So the sequence of partial sums is . . . Non-decreasing Bounded above Geometric

The two “ingredients” together. . . Partial sums are non-decreasing Terms of a series are non-negative Partial sums of are bounded above. and converges

A variant of a familiar theorem Suppose that the sequence is non-decreasing and bounded above by a number A. That is, . . . Theorem 3 on page 553 of OZ Then the series converges to some value that is smaller than or equal to A.

This gives us. . . The Comparison Test: Suppose we have two series , with for all positive integers k. If converges, so does and If diverges, so does .

This test is not in the book! A related test. . . This test is not in the book! There is a test that is closely related to the comparison test, but is generally easier to apply. . . It is called the Limit Comparison Test

(One case of…) The Limit Comparison Test Limit Comparison Test: Consider two series with , each with positive terms. If , then are either both convergent or both divergent. Why does this work?

(Hand waving) Answer: Because if Then for “large” n, ak  t bk.This means that “in the long run” the partial sums behave similarly in terms of convergence or divergence.

The Integral Test y = a(x) Suppose that we have a sequence {ak} and we associate it with a continuous function y = a(x), as we did a few days ago. . . Look at the graph. . . What do you see?

The Integral Test So converges diverges If the integral y = a(x) converges diverges If the integral so does the series.

The Integral Test Now look at this graph. . . What do you see? y = a(x) Now look at this graph. . . What do you see?

The Integral Test So converges diverges If the integral y = a(x) Why 2? converges diverges If the integral so does the series.

The Integral Test The Integral Test: Suppose for all x  1, the function a(x) is continuous, positive, and decreasing. Consider the series and the integral . If the integral converges, then so does the series. If the integral diverges, then so does the series.