Welcome back to Professor Baker's Math Class! In this session, we explored Section 11-3, covering one of the most powerful tools for analyzing infinite series: The Integral Test. We also derived the rules for p-Series and looked at how to estimate the error of a sum.
The Integral Test
The core idea of this section is connecting a discrete series, $\sum a_n$, to a continuous function, $f(x)$. By looking at the area under the curve (the integral), we can determine if the sum of the rectangles (the series) is finite or infinite.
The Conditions: Before applying the test, the function $f(x)$ must satisfy three conditions on the interval $[1, \infty)$:
- Continuous
- Positive
- Decreasing
If these conditions are met, the rule is straightforward:
- If the improper integral $\int_1^{\infty} f(x) \, dx$ is convergent, then the series $\sum_{n=1}^{\infty} a_n$ is convergent.
- If the integral is divergent, then the series is divergent.
Example: Inverse Tangent
In class, we looked at the series $\sum_{n=1}^{\infty} \frac{1}{n^2+1}$. By treating this as the function $f(x) = \frac{1}{x^2+1}$, we evaluated the integral:
$$ \int_1^{\infty} \frac{1}{x^2+1} \, dx = \lim_{t \to \infty} [\tan^{-1}(x)]_1^t = \frac{\pi}{2} - \frac{\pi}{4} = \frac{\pi}{4} $$Since the integral results in a finite number, the series converges.
The p-Series Test
Using the Integral Test, we derived a shortcut known as the p-Series Test for series in the form $\sum_{n=1}^{\infty} \frac{1}{n^p}$. You don't need to integrate these every time; you can simply look at the power $p$:
- If $p > 1$, the series converges.
- If $p \le 1$, the series diverges.
Note: The case where $p=1$ is the famous Harmonic Series, which diverges.
Example: Logarithmic Series
We also analyzed $\sum_{n=1}^{\infty} \frac{\ln n}{n}$. Using $u$-substitution where $u = \ln x$, we found that:
$$ \int_1^{\infty} \frac{\ln x}{x} \, dx = \infty $$Because the integral goes to infinity, this series diverges.
Remainder Estimate
Finally, we discussed how to approximate the sum of a convergent series using the Remainder Estimate. If we sum the first $n$ terms ($S_n$), the error (remainder $R_n$) is bounded by the integral:
$$ \int_{n+1}^{\infty} f(x) \, dx \le R_n \le \int_n^{\infty} f(x) \, dx $$In our class example for $\sum \frac{1}{n^3}$, we found that summing just the first 32 terms is enough to ensure our accuracy is within $0.0005$.
Keep practicing those integrals, and verify your $p$-values! See you in the next class.