Learn to determine series convergence by comparing it to an improper integral. This stack covers the conditions for the Integral Test and its most important application, the p-Series Test.
What are the three conditions a function \(f(x)\) must satisfy to use the Integral Test for a series \(\sum a_n\) where \(a_n = f(n)\)?
For \(x \ge 1\) (or some integer \(N\)), the function \(f(x)\) must be:
1. Continuous
2. Positive
3. Decreasing
If the conditions for the Integral Test are met, what is the conclusion of the test?
The series \(\sum_{n=1}^{\infty} a_n\) converges if and only if the improper integral \(\int_{1}^{\infty} f(x) dx\) converges. Similarly, the series diverges if and only if the integral diverges.
Does the Integral Test tell you the sum of the series?
No. The Integral Test only determines whether the series converges or diverges. The value of the convergent integral is NOT equal to the sum of the series.
Use the Integral Test to determine if \(\sum_{n=2}^{\infty} \frac{1}{n \ln(n)}\) converges or diverges.
Let \(f(x) = \frac{1}{x \ln(x)}\). The integral is \(\int_{2}^{\infty} \frac{1}{x \ln(x)} dx\). Using u-substitution with \(u=\ln(x)\), we get \(\int_{\ln 2}^{\infty} \frac{1}{u} du = [\ln|u|]_{\ln 2}^{\infty} = \infty\). Since the integral diverges, the series also diverges.
What is the general form of a p-series?
A p-series is a series of the form \(\sum_{n=1}^{\infty} \frac{1}{n^p}\), where \(p\) is a positive constant.
State the conclusion of the p-Series Test.
The p-series \(\sum_{n=1}^{\infty} \frac{1}{n^p}\) converges if \(p > 1\) and diverges if \(p \le 1\).
For what values of \(p\) does the p-series \(\sum_{n=1}^{\infty} \frac{1}{n^p}\) converge?
The series converges for \(p > 1\).
For what values of \(p\) does the p-series \(\sum_{n=1}^{\infty} \frac{1}{n^p}\) diverge?
The series diverges for \(p \le 1\).
The p-series with \(p=1\), \(\sum_{n=1}^{\infty} \frac{1}{n}\), is known by what specific name?
It is called the harmonic series, and it diverges.
Does the series \(\sum_{n=1}^{\infty} \frac{1}{\sqrt[3]{n^4}}\) converge or diverge?
This can be rewritten as \(\sum_{n=1}^{\infty} \frac{1}{n^{4/3}}\). This is a p-series with \(p = 4/3\). Since \(p > 1\), the series converges by the p-Series Test.
Does the series \(\sum_{n=1}^{\infty} \frac{\sqrt{n}}{n}\) converge or diverge?
This can be rewritten as \(\sum_{n=1}^{\infty} \frac{n^{1/2}}{n} = \sum_{n=1}^{\infty} \frac{1}{n^{1/2}}\). This is a p-series with \(p = 1/2\). Since \(p \le 1\), the series diverges by the p-Series Test.
The Integral Test Remainder Estimate for \(S = \sum a_n\) is given by \(R_N \le \int_N^{\infty} f(x) dx\). What does \(R_N\) represent?
\(R_N\) represents the error, or remainder, when approximating the total sum \(S\) of the series with the N-th partial sum \(S_N\). Specifically, \(R_N = S - S_N = a_{N+1} + a_{N+2} + \dots\).
