July 01, 2026 - Calculus 2 - Series Convergence: Comparison and Alternating Tests (Concise)

Review of P-Series

• The $P$ series is defined as $\sum_{n=1}^{\infty} \frac{1}{n^p}$.

• Convergence Criteria: The series converges if p > 1 and diverges otherwise ($p \leq 1$).

• The case where $p = 1$ is known as the harmonic series, which is divergent.

The Comparison Test

• Consider two sequences $a_n$ and $b_n$ such that for some integer $N$, $0 \leq a_n \leq b_n$ for all $n \geq N$.

• Part 1 (Convergence): If the series $\sum b_n$ converges, then the series $\sum a_n$ also converges. If the larger series adds up to a finite number, the smaller series must also be finite.

• Part 2 (Divergence): If the series $\sum b_n$ diverges and $a_n \geq b_n$, then the series $\sum a_n$ also diverges. If the smaller series reaches infinity, the larger series must also reach infinity.

• Application Logic:

  • To prove divergence, find a smaller series that is known to diverge.

  • To prove convergence, find a larger series that is known to converge.

The Limit Comparison Test

• This test is used when direct inequalities are difficult to establish, particularly for rational functions.

• Procedure: Take the limit of the ratio of the terms of the series being tested ($a_n$) and a comparison series ($b_n$):     $\lim_{n \to \infty} \frac{a_n}{b_n} = L$

• Case 1: If $L$ is a finite constant and L > 0, then both $\sum a_n$ and $\sum b_n$ either both converge or both diverge.

• Case 2: If $L = 0$ and the series $\sum b_n$ converges, then $\sum a_n$ also converges.

• Case 3: If $L = \infty$ and the series $\sum b_n$ diverges, then $\sum a_n$ also diverges.

• Selection Strategy: For rational functions, compare to a $p$-series using the dominant power of the numerator and denominator (e.g., degree of denominator minus degree of numerator).

The Alternating Series Test

• An alternating series alternates between positive and negative terms, typically written as $\sum (-1)^n b_n$ or $\sum (-1)^{n-1} b_n$, where b_n > 0.

• Conditions for Convergence: A series converges if it satisfies both conditions:

  1. The sequence $b_n$ is decreasing ($b_{n+1} \leq b_n$).

  2. The limit of the sequence is zero: $\lim_{n \to \infty} b_n = 0$.

• If the limit $\lim_{n \to \infty} b_n \neq 0$, the series diverges by the divergence test automatically.

Questions & Discussion

• Question: What was that? We use $N$ whenever we're talking about integers. $C$ would be like any real number.

• Question: So then if it's not a P series that you can say, okay, it converges. Need to find another series to compare it to, right?

  • Response: Yes, if it is not obviously a $p$-series or geometric series, you must find another series to compare it to.

• Question: Is this an alternating series?

  • Response: Yes, though the power of $-1$ only determines if it starts positive or negative.

• Course Logistics: A take-home test will be given on Thursday covering all series. It will be posted online for those attending the FSU orientation. The final two weeks of the course will focus on Power series, Taylor series, and Maclaurin series.