July 02, 2026 - Calculus 2 - Absolute Convergence, Ratio Test, and Root Test Notes (Concise)

Absolute Convergence: A series converges absolutely if the sum of the absolute values of its terms, $\sum |a_n|$ , converges. This distinction is primarily relevant for alternating series.
Conditional Convergence: A series is conditionally convergent if the series $\sum a_n$ converges, but the series of absolute values $\sum |a_n|$ diverges.
Key Example: The harmonic series $\sum \frac{1}{n}$ diverges ( $p$ -series with $p=1$ ), whereas the alternating harmonic series $\sum \frac{(-1)^n}{n}$ converges. Therefore, the alternating harmonic series is conditionally convergent.

An alternating series converges if the following two conditions are satisfied:

The limit of the sequence is zero: $\lim_{n \to \infty} a_n = 0$ .
The terms are non-increasing: $a_{n+1} \le a_n$ .
Application: When evaluating an alternating series for absolute versus conditional convergence, first test for convergence using AST, then test the absolute value of the series using other tests like the $p$ -series test or comparison tests.

The Ratio Test is particularly effective for series involving factorials or exponential growth.
Formulation: Calculate the limit $L = \lim_{n \to \infty} |\frac{a_{n+1}}{a_n}|$ .
Interpretations:
- If L < 1, the series converges absolutely.
- If L > 1 or $L = \infty$ , the series diverges.
- If $L = 1$ , the test is inconclusive, and another test must be utilized.
Factorial Algebra:
- $(n+1)! = (n+1) \times n!$
- $(2n+2)! = (2n+2) \times (2n+1) \times (2n)!$

The Root Test is best suited for series where terms are raised to the power of $n$ .
Formulation: Calculate the limit $L = \lim_{n \to \infty} \sqrt[n]{|a_n|}$ .
Interpretations:
- If L < 1, the series converges absolutely.
- If L > 1 or $L = \infty$ , the series diverges.
- If $L = 1$ , the test is inconclusive.

Euler's Number: A common limit emerging in series testing is $\lim_{n \to \infty} (1 + \frac{1}{n})^n = e$ .
Value of e: Approximately $2.714$ (as noted in the context of ratio comparisons and divergence).
L'Hôpital's Rule: Used when limits result in indeterminate forms such as $\frac{\infty}{\infty}$ .

Q: Why make the distinction between absolute and conditional convergence?
- A: It is critical for power series, which will be discussed in future sessions. Power series only converge in specific intervals based on these conditions.
Q: Is it the same whether a factorial is $n-1$ or $n+1$ ?
- A: You simply adjust the expansion. For example, $n+1$ factorial is $(n+1) \times n!$ . If you have $n$ , you go to $(n) \times (n-1)!$ .
Q: Can you divide the $2$ out of a factorial term?
- A: No. In a term like $(2n)!$ , you are multiplying every number less than $2n$ down to $1$ . Not every individual term in that product is divisible by $2$ .
Q: What happens if the ratio test results in a limit of $1$ ?
- A: The test is inconclusive. For example, applying the ratio test to the alternating harmonic series results in $1$ , even though we know it is conditionally convergent. The test fails to provide information in such cases.