Welcome back to class! Based on our discussions during the last session, I decided it would be beneficial to adjust our pace. Rather than rushing through two sections, we are going to focus exclusively on Section 11-3: The Integral Test and Estimates of Sums. This allows us to really dig into the geometric reasoning behind series convergence without feeling overwhelmed.

The Integral Test

One of the most powerful tools we have for determining if a series converges or diverges is comparing it to an improper integral. As we saw in the lecture slides, looking at the area under a curve $y = f(x)$ can tell us a lot about the sum of the rectangles defined by a series $a_n = f(n)$.

For the Integral Test to apply, the function $f(x)$ must satisfy three specific conditions on the interval $[1, \infty)$:

  • It must be Continuous.
  • It must be Positive.
  • It must be Decreasing.

If these conditions are met, then the following holds true:

$$ \text{If } \int_1^{\infty} f(x) \, dx \text{ is convergent, then } \sum_{n=1}^{\infty} a_n \text{ is convergent.} $$

$$ \text{If } \int_1^{\infty} f(x) \, dx \text{ is divergent, then } \sum_{n=1}^{\infty} a_n \text{ is divergent.} $$

The $p$-Series Test

From the Integral Test, we derive a very useful shortcut known as the $p$-series test. This is crucial for quick analysis of series that look like $\sum \frac{1}{n^p}$. Remember this rule from the notes:

  • The series converges if $p > 1$.
  • The series diverges if $p \le 1$.

For example, $\sum \frac{1}{n^3}$ converges because $3 > 1$, while $\sum \frac{1}{\sqrt{n}}$ diverges because $p = \frac{1}{2} \le 1$.

Estimating Sums

As mentioned in the slides, it is generally difficult to find the exact sum of a series (with rare exceptions like Euler finding that $\sum \frac{1}{n^2} = \frac{\pi^2}{6}$). Therefore, we often rely on partial sums $s_n$ to approximate the total sum $s$.

We can estimate the accuracy of our approximation by looking at the remainder, $R_n = s - s_n$. Using the Remainder Estimate for the Integral Test, we can bound the error:

$$ \int_{n+1}^{\infty} f(x) \, dx \le R_n \le \int_n^{\infty} f(x) \, dx $$

This inequality allows us to determine how many terms ($n$) we need to add up to ensure our calculation is within a specific margin of error, such as $0.0005$.

Be sure to review the annotated slides attached below, specifically Example 4 regarding the derivative test for decreasing functions and Example 5 for error estimation.

Class Materials:
Zoom Meeting Link | Notes 11-3 PPT | Notes 11-3 PDF | Class Notes