Welcome back to our exploration of Infinite Sequences and Series! In this lesson, we tackle Sections 11.3 and 11.4, moving beyond the basic definition of convergence to specific, powerful tests that help us determine if a series sums up to a finite number or grows to infinity.

Below, you will find the essential resources for tonight's lecture, followed by a breakdown of the key concepts we are covering.

11.3 The Integral Test & Estimates of Sums

Finding the exact sum of a series is often difficult. However, determining whether it converges is manageable by relating the sum to an improper integral. We can visualize the terms of a series as rectangles under a curve. If the area under the curve is finite, the sum of the rectangles must be finite too.

The Integral Test:
Suppose $f$ is a continuous, positive, and decreasing function on $[1, \infty)$ and let $a_n = f(n)$. Then the series $\sum_{n=1}^{\infty} a_n$ is convergent if and only if the improper integral is convergent:

$$\int_1^{\infty} f(x) \, dx$$

If the integral diverges, the series diverges.

The p-series Test

From the Integral Test, we derive one of the most useful benchmarks in this chapter: the p-series. A p-series is given by:

$$\sum_{n=1}^{\infty} \frac{1}{n^p}$$
  • If $p > 1$, the series converges.
  • If $p \le 1$, the series diverges.

11.4 The Comparison Tests

Sometimes, integrating a function is too complicated. In these cases, we use Comparison Tests. The strategy is to compare a messy, unknown series to a simple, known benchmark (usually a p-series or a geometric series).

1. The Direct Comparison Test

Suppose we have two series with positive terms, $\sum a_n$ and $\sum b_n$.

  • If $\sum b_n$ is convergent and $a_n \le b_n$, then $\sum a_n$ is also convergent. (If the "larger" series fits, the "smaller" one must also fit).
  • If $\sum b_n$ is divergent and $a_n \ge b_n$, then $\sum a_n$ is also divergent.

2. The Limit Comparison Test

Often, inequalities are hard to prove directly. The Limit Comparison Test is frequently easier to apply. We calculate the limit of the ratio of the two series:

$$ \lim_{n \to \infty} \frac{a_n}{b_n} = c $$

If $c$ is a finite number and $c > 0$, then both series behave the same way—either both converge or both diverge. This allows us to ignore "less important" terms in complex fractions and focus on the dominant powers.

Remember to review the attached PDF for examples of how to apply these tests to logarithmic and algebraic functions!