Infinite series can add up to a finite value even when they contain infinitely many terms. In calculus, convergence tells us whether the partial sums settle toward a limit, while divergence means they do not. Absolute and conditional convergence describe two different ways an infinite series can converge.
The distinction matters because it tells us how stable the sum is under changes such as rearranging the terms.
To test absolute convergence, replace every term a_n by its magnitude |a_n| and study the positive series sum |a_n|. If sum |a_n| converges, then sum a_n converges absolutely and is very robust. If sum a_n converges but sum |a_n| diverges, then the series is conditionally convergent and its value depends on the order of the terms.
Alternating series such as sum (-1)^(n+1)/n are classic examples because cancellation can create convergence even when the total size of the terms is too large.
Key Facts
- A series sum a_n converges if its partial sums S_N = a_1 + a_2 + ... + a_N approach a finite limit.
- A series sum a_n converges absolutely if sum |a_n| converges.
- Absolute convergence implies ordinary convergence: if sum |a_n| converges, then sum a_n converges.
- A series sum a_n converges conditionally if sum a_n converges but sum |a_n| diverges.
- Alternating Series Test: sum (-1)^(n+1)b_n converges if b_n > 0, b_n decreases, and lim n to infinity b_n = 0.
- p-series rule: sum 1/n^p converges if p > 1 and diverges if p <= 1.
Vocabulary
- Infinite series
- An infinite series is a sum of infinitely many terms, written as sum a_n.
- Partial sum
- A partial sum is the finite sum S_N = a_1 + a_2 + ... + a_N used to approximate an infinite series.
- Absolute convergence
- Absolute convergence occurs when the series of absolute values sum |a_n| converges.
- Conditional convergence
- Conditional convergence occurs when sum a_n converges but sum |a_n| diverges.
- Alternating series
- An alternating series is a series whose terms switch signs, often written in the form sum (-1)^n b_n or sum (-1)^(n+1)b_n.
Common Mistakes to Avoid
- Assuming convergence of sum a_n means absolute convergence. This is wrong because a series can converge only through cancellation while sum |a_n| still diverges.
- Forgetting to test the absolute value series first. This is wrong because sum |a_n| gives the strongest conclusion, and if it converges then no separate conditional test is needed.
- Using the Alternating Series Test when b_n does not decrease to zero. This is wrong because the test requires positive terms b_n that eventually decrease and have limit 0.
- Thinking a conditionally convergent series has a fixed sum no matter how terms are rearranged. This is wrong because rearranging a conditionally convergent series can change its sum or even make it diverge.
Practice Questions
- 1 Determine whether sum from n = 1 to infinity of (-1)^(n+1)/n is absolutely convergent, conditionally convergent, or divergent.
- 2 Determine whether sum from n = 1 to infinity of (-1)^n/n^2 is absolutely convergent, conditionally convergent, or divergent.
- 3 Explain why absolute convergence is considered more stable than conditional convergence when the terms of a series are rearranged.