Source: Integral Calculus, University of Texas at Austin
Tags: absolute convergence, conditional convergence, alternating series, comparison test, geometric series, factorial growth, Calculus II
Absolute convergence means a series still converges even when you take the absolute value of every term. Conditional convergence means it only converges because of sign cancellation. The distinction matters because absolutely convergent series are better behaved (you can rearrange terms freely). This set of problems trains you to classify series into one of three buckets: absolutely convergent, conditionally convergent, or divergent.
Difficulty: Intermediate
Prerequisites: alternating series test, comparison test, geometric series, p-series, basic understanding of factorials.
Once you can decide whether a series converges, the next question is how it converges. Absolute convergence is the stronger property: the series converges even without any helpful sign cancellation. Conditional convergence is the weaker one, where the series converges only because positive and negative terms partially cancel each other out.
This classification shows up again when you study power series (the interior of the interval of convergence is absolute, the endpoints may be conditional) and in rearrangement theorems (conditionally convergent series can be rearranged to sum to any value, a result that surprises most students).
Absolutely convergent
A series sum of a_n is absolutely convergent if sum of |a_n| converges. Think of it as: the series converges "on its own strength," without needing sign cancellation to help.
Conditionally convergent
A series that converges, but whose absolute-value version diverges. In simple terms, it only converges because the positive and negative terms keep pulling the partial sums back toward a limit.
Geometric series
The series sum of r^n. Converges when |r| < 1 (to 1/(1-r) starting from n=0). This is the comparison target whenever your terms involve exponentials in the denominator (like 3^n).
Factorial growth
Factorials grow faster than any exponential: n! eventually outpaces c^n for any constant c. This means series with n! in the denominator tend to converge, and those with n! in the numerator tend to diverge.
Comparison Test
If 0 <= a_n <= b_n for all n (eventually), then: if sum b_n converges, so does sum a_n; if sum a_n diverges, so does sum b_n. Think of it as: a series smaller than a convergent series must converge; a series larger than a divergent series must diverge.
When a problem says "determine whether the series is absolutely convergent, conditionally convergent, or divergent," follow this order:
Test for absolute convergence first: take the absolute value of the terms and check whether that new (positive) series converges.
If the absolute-value series converges, you are done: the original series is absolutely convergent.
If the absolute-value series diverges, test whether the original series converges (typically using the AST for alternating series).
If the original converges but the absolute version diverges, it is conditionally convergent.
If neither converges, the series is divergent.
The series sum of sin(8n) / 3^n: take absolute values. Since |sin(8n)| <= 1, we have |sin(8n) / 3^n| <= 1/3^n. The series sum of 1/3^n is a convergent geometric series (r = 1/3 < 1). By the Comparison Test, sum of |sin(8n)/3^n| converges. So the original series is absolutely convergent.
This pattern applies broadly: whenever the numerator is bounded (sines, cosines, bounded sequences) and the denominator grows exponentially, the series is absolutely convergent by comparison with a geometric series.
The series sum of (7 * 14 * 21 * ... * 7n) / n!: the numerator is 7^n * n!, so the terms simplify to 7^n * n! / n! = 7^n. Since 7^n grows without bound, the terms do not approach zero. The series diverges by the Divergence Test.
More carefully, the numerator is the product 7 * 14 * 21 * ... * 7n = 7^n * (1 * 2 * 3 * ... * n) = 7^n * n!. Dividing by n! gives 7^n, and since lim 7^n = infinity, the series diverges.
The series sum of (-1)^(n+1) * e^(1/n) / (3n):
Absolute convergence: |a_n| = e^(1/n) / (3n). As n gets large, e^(1/n) approaches e^0 = 1, so |a_n| behaves like 1/(3n). This is a constant multiple of the harmonic series, which diverges. So the series is not absolutely convergent.
Conditional convergence: the series is alternating with b_n = e^(1/n) / (3n). Since e^(1/n) approaches 1 and 1/(3n) decreases to 0, b_n is eventually decreasing and approaches 0. By the AST, the series converges.
Verdict: conditionally convergent.
This is sum from n=4 of (-1)^n * 3/n (after reindexing). The absolute-value series is sum of 3/n (starting at n=4), a constant multiple of the harmonic series, which diverges. But the alternating version satisfies the AST (3/n is decreasing and approaches 0). Conditionally convergent.
The series sum of (-5)^n / (2n)!: take absolute values to get sum of 5^n / (2n)!. Factorials grow much faster than exponentials, so 5^n / (2n)! approaches 0 extremely fast. The Ratio Test confirms convergence (the ratio of consecutive terms approaches 0). Absolutely convergent.
This is a useful pattern to remember: any series of the form c^n / (kn)! converges absolutely, because factorial growth in the denominator dominates.
Absolute convergence criterion
If sum of |a_n| converges, then sum of a_n converges (absolutely).
Geometric series
sum of r^n (n=0 to infinity) = 1/(1-r) when |r| < 1. Diverges when |r| >= 1.
Comparison Test (for absolute convergence)
If |a_n| <= c * r^n for some c > 0 and |r| < 1, then sum of a_n is absolutely convergent.
Useful bound
|sin(x)| <= 1 and |cos(x)| <= 1 for all x. This lets you replace oscillating numerators with a constant bound.
Students often think "conditionally convergent" means "barely converges" or "might not converge." It does converge, with certainty. The word "conditional" refers to the fact that convergence depends on the sign pattern, not that it is uncertain.
A frequent mistake is testing for conditional convergence first. Always test absolute convergence first. If the series is absolutely convergent, you are done and do not need the AST.
Students sometimes forget that |sin(n)| is bounded by 1 and try to evaluate it exactly. For convergence questions, the bound is all you need.
When a product like 7 * 14 * 21 * ... * 7n appears, students sometimes fail to recognise it as 7^n * n!. Factoring out the common ratio from each term reveals the structure.
⚠️ "Absolutely convergent, conditionally convergent, or divergent" is a three-way classification. Exams expect you to rule out absolute convergence before concluding conditional convergence.
⚠️ Comparison with geometric series (sum of 1/r^n) is the most common route to proving absolute convergence when there is an exponential in the denominator.
⚠️ Factorial denominators are your friend. Any series with (2n)! or n! in the denominator and only exponentials or polynomials in the numerator will almost certainly converge absolutely.
⚠️ The series 3/(n+3) pattern (harmonic-type starting late) still diverges. Shifting the starting index does not affect convergence.
True or False: If a series is absolutely convergent, it is also convergent.
True. Absolute convergence implies convergence.
True or False: If a series is conditionally convergent, the series of absolute values converges.
False. That is exactly what "conditional" means: the absolute-value series diverges.
Fill in the blank: |sin(8n)| <= ______, which lets us compare sum of sin(8n)/3^n with a geometric series.
True or False: The series sum of 7^n diverges.
True. The terms grow without bound.
Q: Is sum of sin(8n)/3^n absolutely convergent, conditionally convergent, or divergent?
A: Absolutely convergent. |sin(8n)/3^n| <= 1/3^n, and sum of 1/3^n is a convergent geometric series.
Q: Is sum of (-1)^(n+1) * e^(1/n) / (3n) absolutely convergent, conditionally convergent, or divergent?
A: Conditionally convergent. The absolute-value series behaves like 1/(3n), which diverges. But the alternating series passes the AST.
Q: Is sum of (-5)^n / (2n)! absolutely convergent, conditionally convergent, or divergent?
A: Absolutely convergent. sum of 5^n / (2n)! converges because factorial growth dominates exponential growth.
Q: The series 3/4 - 3/5 + 3/6 - 3/7 + ... is conditionally convergent, absolutely convergent, or divergent?
A: Conditionally convergent. The absolute-value version is sum of 3/n (from n=4), which diverges. The alternating version converges by the AST.
Q: Is sum of (7 * 14 * 21 * ... * 7n) / n! absolutely convergent, conditionally convergent, or divergent?
A: Divergent. The terms simplify to 7^n, which does not approach zero.
Absolute convergence connects to the Ratio Test and Root Test (covered in the next set of notes), which are the primary tools for proving absolute convergence of series involving factorials and exponentials. It also connects to power series: inside the radius of convergence, a power series converges absolutely; at the endpoints, it may converge conditionally or diverge.
The Comparison Test used here reappears throughout real analysis and is one of the foundational ideas in measure theory and integration.
Absolute convergence, conditional convergence, comparison test, geometric series, factorial growth, alternating series, Leibniz test, divergence test, bounded numerator, sin(n)/3^n, (2n)!, Calculus II, integral calculus, UT Austin, HW 11, series classification