10.7 Alternating Series Test for Convergence

A series is alternating if its terms switch sign every term (plus, minus, plus, minus…). Practically you’ll see this as a factor like (−1)^n or (−1)^{n+1} multiplying a positive-term sequence a_n. Examples: ∑ (−1)^n(1/n) and ∑ (−1)^{n+1} e^{-n} are alternating; ∑ 1/n is not. How to check quickly: - Look for an explicit (−1)^n or (−1)^{n+1} factor. - If signs shown term-by-term alternate (+, −, +, −…), it’s alternating. - If unsure, rewrite term t_n = (−1)^n a_n and verify a_n ≥ 0 for all n. Why it matters for the AP Alternating Series Test (LIM-7.A.10): if the series is alternating and the a_n are monotonically decreasing to 0, the series converges (possibly only conditionally). To check absolute vs conditional convergence, test ∑ |t_n| separately. For more examples and step-by-step practice aligned to the CED, see the Topic 10.7 study guide (https://library.fiveable.me/ap-calculus/unit-10/alternating-series-test-for-convergence/study-guide/XSi4mU7Hy7oj9k5cIdqe). For broader unit review and lots of practice questions, visit (https://library.fiveable.me/ap-calculus/unit-10) and (https://library.fiveable.me/practice/ap-calculus).

If you’ve got an alternating series ∑ (-1)^{n} a_n (or with (-1)^{n+1}), the Alternating Series Test (Leibniz criterion) says the series converges if two conditions hold: 1) a_n ≥ 0 and a_{n+1} ≤ a_n for all large n (the terms decrease monotonically in absolute value), and 2) lim_{n→∞} a_n = 0. So formally: if a_n ↓ 0, then ∑_{n=1}^∞ (-1)^{n} a_n converges. Also useful on the exam: the Alternating Series Estimation Theorem gives an error bound for partial sums—the remainder after N terms satisfies |R_N| ≤ a_{N+1} (the next term’s absolute value). This is a BC-only topic in Unit 10 (CED LIM-7.A.10). For a quick study guide, check the Unit 10 topic page (https://library.fiveable.me/ap-calculus/unit-10/alternating-series-test-for-convergence/study-guide/XSi4mU7Hy7oj9k5cIdqe). For extra practice, see the AP Calculus practice bank (https://library.fiveable.me/practice/ap-calculus).

Use the Alternating Series Test (Leibniz criterion) when the series has strictly alternating signs (terms like (−1)^n a_n or similar) and you can check two things: (1) the sequence of absolute terms a_n is eventually monotonically decreasing, and (2) a_n → 0. If both hold, the series converges (this is the AST—LIM-7.A.10 in the CED) and you can also use the Alternating Series Estimation Theorem to bound the error by the next term. If the series isn’t alternating, or the absolute terms don’t decrease to 0, use other tests: - nth-term test first (if a_n ↛ 0 the series diverges). - Ratio or Root Test for factorials/exponentials (or when terms involve n! or r^n). - Comparison / Limit Comparison or Integral Test / p-series when terms are positive and behave like 1/n^p. - If the series of |a_n| converges, you get absolute convergence (stronger than AST’s conditional convergence). On the AP exam you should explicitly verify the AST conditions (monotone decrease and limit zero) when applying it (see Topic 10.7 study guide for examples: https://library.fiveable.me/ap-calculus/unit-10/alternating-series-test-for-convergence/study-guide/XSi4mU7Hy7oj9k5cIdqe). For extra practice across Unit 10, try problems at https://library.fiveable.me/practice/ap-calculus or review the whole unit overview (https://library.fiveable.me/ap-calculus/unit-10).

The Alternating Series Test (Leibniz criterion) checks an alternating series ∑(-1)^{k} a_k (with a_k ≥ 0) in three quick steps: 1. Identify the positive-term sequence a_k. Write the series as +a_1 - a_2 + a_3 - … so a_k ≥ 0. 2. Check monotonic decrease: verify a_{k+1} ≤ a_k for all large k (eventually nonincreasing). This often uses calculus (take derivative of a continuous version) or simple algebra. 3. Check the termwise limit: lim_{k→∞} a_k = 0. If both (2) and (3) hold, the series converges (may be only conditionally—not necessarily absolutely). If lim a_k ≠ 0, it diverges. To test absolute convergence, check ∑ a_k separately (if that converges, the original converges absolutely). Error bound: the Alternating Series Estimation Theorem says the truncation error after N terms is ≤ a_{N+1} (useful on the exam to justify approximations). For AP BC review, see the Topic 10.7 study guide (https://library.fiveable.me/ap-calculus/unit-10/alternating-series-test-for-convergence/study-guide/XSi4mU7Hy7oj9k5cIdqe) and extra practice (https://library.fiveable.me/practice/ap-calculus).

No—an alternating series just means the signs switch each term, not that it must start negative. Typical forms are ∑(-1)^{n} a_n (first term negative) or ∑(-1)^{n+1} a_n (first term positive). What matters for the Alternating Series Test (Leibniz criterion) is that the sequence of absolute term sizes a_n is positive, eventually monotonically decreasing, and a_n → 0. If those hold, the alternating series converges (possibly only conditionally). Example: the alternating harmonic ∑(-1)^{n+1}/n starts positive (1 − 1/2 + 1/3 − …) and converges; ∑(-1)^n/n is the same series shifted and starts negative (−1 + 1/2 − 1/3 + …). Also remember the Alternating Series Estimation Theorem: the error from truncating is at most the next term’s absolute value. For AP review, see the Topic 10.7 study guide (https://library.fiveable.me/ap-calculus/unit-10/alternating-series-test-for-convergence/study-guide/XSi4mU7Hy7oj9k5cIdqe) and more practice problems (https://library.fiveable.me/practice/ap-calculus).

Do these steps every time you see an alternating series: 1. Identify the sign pattern (terms like (−1)^n or (−1)^{n+1} mean alternating). 2. Check the term magnitudes an = |a_n|: are they eventually monotonically decreasing? (If not, the test may not apply.) 3. Check the limit: lim_{n→∞} an = 0. If it’s not 0, the series diverges by the nth-term test. 4. If an decreases to 0, the Alternating Series Test (Leibniz criterion) says the series converges. This may be conditional convergence—test absolute convergence by checking ∑|a_n| (if that converges, the original is absolutely convergent). 5. Error bound: the Alternating Series Estimation Theorem says the remainder after N terms is ≤ a_{N+1}. Use that to get approximations. Quick example: ∑ (−1)^{n+1}/n has decreasing terms with limit 0, so it converges (but ∑1/n diverges, so it’s conditionally convergent). For an AP-aligned review, see the Topic 10.7 study guide (https://library.fiveable.me/ap-calculus/unit-10/alternating-series-test-for-convergence/study-guide/XSi4mU7Hy7oj9k5cIdqe) and try practice problems (https://library.fiveable.me/practice/ap-calculus).

Absolute vs. conditional convergence for alternating series is about what happens when you drop the signs. - Absolute convergence: The series ∑ a_n (with signs) converges and the series of absolute values ∑ |a_n| also converges. If ∑ |a_n| converges, then ∑ a_n converges too. Absolute convergence is stronger and rules out weird rearrangement effects. - Conditional convergence: The alternating series ∑ (-1)^n b_n converges (usually by the Alternating Series Test: b_n ≥ 0, b_n decreasing, and b_n → 0) but ∑ |(-1)^n b_n| = ∑ b_n diverges. Classic example: alternating harmonic ∑ (-1)^{n+1}/n converges by the Alternating Series Test but its absolute series (the harmonic series) diverges—so it’s conditionally convergent. On the AP BC exam, you should be able to apply the Alternating Series Test (LIM-7.A.10) and check absolute convergence by testing ∑ |a_n|. For a quick review, see the Topic 10.7 study guide (https://library.fiveable.me/ap-calculus/unit-10/alternating-series-test-for-convergence/study-guide/XSi4mU7Hy7oj9k5cIdqe). For extra practice, try problems at (https://library.fiveable.me/practice/ap-calculus).

Short answer: yes—you need the absolute values of the terms (call them b_n = |a_n| for an alternating series a_n = (−1)^{n} b_n) to be eventually decreasing and to have limit 0 for the Alternating Series Test (Leibniz criterion) to guarantee convergence. More detail: AST says if b_n ≥ 0, b_{n+1} ≤ b_n for all large n (i.e., eventually monotonically decreasing), and lim_{n→∞} b_n = 0, then the alternating series converges (possibly conditionally). If b_n isn’t monotone at first but becomes monotone after some index N, that’s fine—you can start the test at N. If lim b_n ≠ 0 the series diverges. Also remember AST doesn’t test absolute convergence: check ∑|a_n| separately if you need absolute vs conditional convergence. The Alternating Series Estimation Theorem also gives an error bound: the truncation error ≤ next term b_{n+1}. For more review and examples tied to the AP CED (LIM-7.A), see the Topic 10.7 study guide (https://library.fiveable.me/ap-calculus/unit-10/alternating-series-test-for-convergence/study-guide/XSi4mU7Hy7oj9k5cIdqe) and try practice problems (https://library.fiveable.me/practice/ap-calculus).

Other tests (like ratio, root, p-series) often look at size of terms only. The Alternating Series Test works when signs alternate because the signs cause cancellation: if a_n ≥ 0, a_n is decreasing, and a_n → 0, then partial sums form two monotone sequences (even and odd partial sums) that squeeze toward the same limit—so the series converges. That’s Leibniz’s criterion in the CED (LIM-7.A.10). Importantly, the test can show conditional convergence: the series converges even though the series of absolute values diverges. The Alternating Series Estimation Theorem also gives a simple error bound: the truncation error ≤ the next term’s absolute value. That’s why AST succeeds where size-only tests fail—alternating signs create cancellation and a built-in error control. For more practice and the AP-aligned study guide, see the Topic 10.7 study guide (https://library.fiveable.me/ap-calculus/unit-10/alternating-series-test-for-convergence/study-guide/XSi4mU7Hy7oj9k5cIdqe) and Unit 10 overview (https://library.fiveable.me/ap-calculus/unit-10). For extra problems, try the practice bank (https://library.fiveable.me/practice/ap-calculus).

If an alternating series “fails” the Alternating Series Test (AST), what that means depends on which AST condition fails: - If the termwise limit doesn’t go to 0 (lim a_n ≠ 0), the series diverges immediately by the nth-term test. - If the terms do go to 0 but they aren’t eventually monotone decreasing, the AST is inconclusive: the series might still converge (often conditionally) or might diverge. You can’t conclude either way from AST alone. - If you want a definitive answer when AST fails, try other tests: absolute-convergence tests (compare, ratio, root) to check for absolute convergence; conditional-convergence tests (Dirichlet/Abel) or grouping/rearrangement arguments where appropriate. Remember on the AP BC exam (CED LIM-7.A) you should identify which AST condition fails and either use the nth-term test (for nonzero limits) or apply another convergence test. For a quick review, see the Topic 10.7 study guide (https://library.fiveable.me/ap-calculus/unit-10/alternating-series-test-for-convergence/study-guide/XSi4mU7Hy7oj9k5cIdqe) and try related practice problems (https://library.fiveable.me/practice/ap-calculus).

Use the Alternating Series Test (Leibniz) when the series is alternating in sign and you can show two things: the absolute values of the terms are eventually monotonically decreasing, and the terms go to 0. If both hold, the series converges (often only conditionally). You can also use the Alternating Series Estimation Theorem to bound the remainder by the next term. Use the Ratio Test when terms involve factorials, exponentials, powers, or products where |a_{n+1}/a_n| simplifies nicely. The Ratio Test tells you about absolute convergence: if L = lim |a_{n+1}/a_n| < 1 the series converges absolutely; if L > 1 it diverges; if L = 1 it's inconclusive. Practical checklist: first look at form. If signs alternate and terms are simple decreasing to 0 → try Alternating Series Test. If factorials/exponentials/powers or you want absolute convergence → try Ratio Test. If Ratio Test gives L = 1 but signs alternate, fall back to the Alternating Series Test. For more AP-aligned review see the Topic 10.7 study guide (https://library.fiveable.me/ap-calculus/unit-10/alternating-series-test-for-convergence/study-guide/XSi4mU7Hy7oj9k5cIdqe) and unit resources (https://library.fiveable.me/ap-calculus/unit-10). Practice problems are at (https://library.fiveable.me/practice/ap-calculus).

Think of (-1)^n as a sign-flipper. For n = 1, 2, 3, 4, ... the values of (-1)^n are -1, 1, -1, 1, ... (or if you start at n=0 you get 1, -1, 1, -1, ...). Multiplying a positive-term sequence a_n by (-1)^n makes the signs alternate +, −, +, − (or the opposite), so the terms of the series change sign every term—that’s exactly what “alternating series” means. Why this matters for convergence: the Alternating Series Test (Leibniz criterion) says a series ∑ (-1)^n a_n converges if a_n ≥ 0, a_n is eventually monotonically decreasing, and lim_{n→∞} a_n = 0. If those conditions hold, the series converges (maybe conditionally), and the Alternating Series Estimation Theorem gives an error bound by the next term. For more AP-aligned review, see the Topic 10.7 study guide (https://library.fiveable.me/ap-calculus/unit-10/alternating-series-test-for-convergence/study-guide/XSi4mU7Hy7oj9k5cIdqe). For extra practice problems, check Fiveable’s practice page (https://library.fiveable.me/practice/ap-calculus).

You check two things for the Alternating Series Test (Leibniz criterion): 1. The absolute values of the terms, b_n = |a_n|, are eventually monotonically decreasing: b_{n+1} ≤ b_n for all n large enough. 2. The terms go to zero: lim_{n→∞} b_n = 0. If both hold for an alternating series ∑ (−1)^n b_n (or (−1)^{n+1} b_n), the series converges (possibly only conditionally). Note: if ∑ b_n also converges, the series converges absolutely. For error control, the Alternating Series Estimation Theorem says the error after N terms is ≤ next term b_{N+1}. This is exactly the LIM-7.A knowledge in the CED. For a short study guide and examples, see the Topic 10.7 study guide (https://library.fiveable.me/ap-calculus/unit-10/alternating-series-test-for-convergence/study-guide/XSi4mU7Hy7oj9k5cIdqe). For lots of practice problems, check (https://library.fiveable.me/practice/ap-calculus).

Yes—an alternating series can converge even when the corresponding non-alternating (absolute) series diverges. That’s called conditional convergence. Key points (AP CED keywords): - Alternating Series Test (Leibniz criterion): if a_n ≥ 0, a_{n+1} ≤ a_n (eventually) and a_n → 0, then sum (-1)^{n} a_n converges. - But absolute convergence means sum |(-1)^n a_n| = sum a_n converges. If the alternating series converges but sum a_n diverges, the series is conditionally convergent. Classic example: the alternating harmonic series sum_{n=1}^∞ (-1)^{n+1} (1/n) converges by the Alternating Series Test, while the harmonic series sum 1/n diverges—so it’s conditionally convergent. On the exam: be ready to check monotone decreasing terms and limit zero for the Alternating Series Test, and distinguish conditional vs absolute convergence. For a refresher, see the Topic 10.7 study guide (https://library.fiveable.me/ap-calculus/unit-10/alternating-series-test-for-convergence/study-guide/XSi4mU7Hy7oj9k5cIdqe) and practice problems (https://library.fiveable.me/practice/ap-calculus).

Keep it short and formulaic—graders want the test, why it applies, and a clear conclusion. 1) State the series and note it’s alternating (shows (-1)^n or sign flip). 2) List the two AST conditions: (i) the absolute terms b_n are eventually positive and decreasing (b_{n+1} ≤ b_n) and (ii) lim_{n→∞} b_n = 0. Cite “Alternating Series Test (Leibniz criterion).” (CED: LIM-7.A.10) 3) Verify each condition with a short justification (derivative or simple inequality for monotonicity; limit evaluation for →0). If either fails, say diverges by AST. If both hold, conclude the series converges (conditional unless you also check absolute convergence). 4) If asked for an error bound / estimate, use the Alternating Series Estimation Theorem: remainder ≤ next term b_{N+1} and give that numeric bound. 5) Wrap up: state convergence type (converges conditionally or absolutely). For examples and practice problems, see the Topic 10.7 study guide (https://library.fiveable.me/ap-calculus/unit-10/alternating-series-test-for-convergence/study-guide/XSi4mU7Hy7oj9k5cIdqe) and extra practice (https://library.fiveable.me/practice/ap-calculus).