10.3 The nth Term Test for Divergence

You use the nth-term (zero) test to quickly rule out convergence. It says: if lim_{n→∞} a_n ≠ 0 or does not exist, then the series ∑ a_n diverges. So first compute L = lim_{n→∞} a_n. - If L ≠ 0 (for example a_n = 5 for all n, or a_n = (−1)^n), the series diverges. - If L = 0, the test is inconclusive—the series might converge (like a geometric series with |r|<1) or diverge (the harmonic series ∑1/n has terms →0 but still diverges). Remember: the nth-term test is a necessary condition for convergence, not sufficient. On the AP BC exam you’ll use it to eliminate options fast (CED LIM-7.A.5). For a short guide and examples, see the Topic 10.3 study guide (https://library.fiveable.me/ap-calculus/unit-10/nth-term-test-for-divergence/study-guide/oEMEbEp7gWXCgxDFyRlx). Practice lots of problems at (https://library.fiveable.me/practice/ap-calculus) to get fluent.

The nth-term (or zero) test says: if the sequence of terms a_n does not approach 0, the series ∑ a_n must diverge. In formula form: - If lim_{n→∞} a_n ≠ 0 or the limit does not exist, then ∑_{n=1}^∞ a_n diverges. - If lim_{n→∞} a_n = 0, the test is inconclusive (the series may converge or diverge). So the necessary condition for convergence is lim a_n = 0, but that alone isn’t enough (e.g., the harmonic series has terms →0 but diverges; a geometric series with |r|<1 converges). This is the very first divergence test you should check on AP BC series problems (CED LIM-7.A.5). For a short study guide on this topic, see the Topic 10.3 study guide (https://library.fiveable.me/ap-calculus/unit-10/nth-term-test-for-divergence/study-guide/oEMEbEp7gWXCgxDFyRlx). For more practice, use the AP Calculus practice bank (https://library.fiveable.me/practice/ap-calculus).

Use the nth-term (zero) test first as a quick fail-fast check: if lim_{n→∞} a_n ≠ 0 (or doesn’t exist), the series ∑a_n diverges—that’s all the test can prove (CED: LIM-7.A.5). If lim a_n = 0, the test is inconclusive—the series may converge or diverge. When to move to other tests: - If a_n ≈ geometric form → use geometric-series test. - If a_n ~ 1/n^p → use p-series (or comparison). - If terms are positive and come from a nice function → try the integral test. - If you can bound a_n above/below by a known series → comparison or limit comparison. - If terms alternate and decrease to 0 → alternating series test (for conditional convergence). - If factorials/exponentials appear → ratio test. - If partial sums simplify → telescoping. Strategy for AP BC: always do the nth-term test first (fast check). If it’s inconclusive, pick the test that matches the term’s form. For a refresher, see the Topic 10.3 study guide (https://library.fiveable.me/ap-calculus/unit-10/nth-term-test-for-divergence/study-guide/oEMEbEp7gWXCgxDFyRlx) and practice problems (https://library.fiveable.me/practice/ap-calculus).

No—lim_{n→∞} a_n = 0 is necessary for ∑ a_n to converge, but it’s not sufficient. That’s exactly what the nth-term (divergence) test says: if lim_{n→∞} a_n ≠ 0 (or doesn’t exist), then the series diverges. But if lim_{n→∞} a_n = 0, the test is inconclusive—the series may converge or diverge. Example: a_n = 1/n has lim a_n = 0, but the harmonic series ∑ 1/n diverges. Contrast that with the alternating harmonic series ∑ (-1)^{n+1}/n, which converges (conditionally). So when terms go to 0 you must use other tests (p-series, comparison, integral, alternating series, ratio, etc.) to decide convergence—this is part of LIM-7.A in the CED. For a quick refresher on the divergence test, see the Topic 10.3 study guide (https://library.fiveable.me/ap-calculus/unit-10/nth-term-test-for-divergence/study-guide/oEMEbEp7gWXCgxDFyRlx). For more practice problems across Unit 10, check the unit page (https://library.fiveable.me/ap-calculus/unit-10) or the practice bank (https://library.fiveable.me/practice/ap-calculus).

The nth-term test only proves divergence. AP CED (LIM-7.A.5) calls it a divergence test: if lim_{n→∞} a_n ≠ 0 (or the limit doesn’t exist), then the series ∑ a_n must diverge. That’s a necessary condition for convergence—a_n → 0 is required but not enough. So: - If lim a_n ≠ 0 → series diverges (you can conclude divergence). - If lim a_n = 0 → the test is inconclusive; the series might converge (e.g., geometric with |r|<1) or diverge (e.g., harmonic ∑1/n). For more AP-aligned notes and examples, check the Topic 10.3 study guide (https://library.fiveable.me/ap-calculus/unit-10/nth-term-test-for-divergence/study-guide/oEMEbEp7gWXCgxDFyRlx). For extra practice problems across Unit 10, use (https://library.fiveable.me/practice/ap-calculus).

Use the nth-term (zero) test as a quick divergence check—step by step: 1. Identify a_n, the general term of the series ∑ a_n. 2. Compute L = lim_{n→∞} a_n. (This is a limit of the sequence of terms, not partial sums.) 3. Apply the rule (CED/LIM-7.A.5): - If L ≠ 0 or the limit doesn’t exist, the series diverges. (Necessary condition for convergence: terms must go to 0.) - If L = 0, the test is inconclusive—the series might converge or diverge; use other tests (comparison, integral, alternating, ratio, etc.). 4. Common checks: for a_n = 1/n, L = 0 but harmonic series still diverges (so nth-term test alone fails); for a_n = 1/2^n, L = 0 and geometric test shows convergence. So practically: take the limit; nonzero → immediate “diverges.” Zero → stop and pick a stronger convergence test. For the AP BC topic and examples, see the Fiveable study guide (https://library.fiveable.me/ap-calculus/unit-10/nth-term-test-for-divergence/study-guide/oEMEbEp7gWXCgxDFyRlx). For extra practice, use the Unit 10 overview (https://library.fiveable.me/ap-calculus/unit-10) and Fiveable’s practice problems (https://library.fiveable.me/practice/ap-calculus).

The nth-term (divergence) test and the ratio test do related but different jobs. - nth-term test (zero test): If lim_{n→∞} a_n ≠ 0 (or doesn’t exist) then the series ∑ a_n diverges. It’s a necessary condition for convergence only—if lim a_n = 0 you still don’t know convergence. This is the simple “nonzero term ⇒ divergence” test emphasized in LIM-7.A (Topic 10.3) and is the first quick check you should always do. - Ratio test: Compute L = lim_{n→∞} |a_{n+1}/a_n|. - If L < 1, the series converges absolutely. - If L > 1 (or = ∞), it diverges. - If L = 1, the test is inconclusive. Use the ratio test for factorials, exponentials, or n-th powers (e.g., n!, 2^n, (3/2)^n). It often succeeds where the nth-term test can’t (because nth-term only rules out convergence). Quick strategy for AP BC: always run the nth-term test first (zero test); if lim a_n = 0, try a stronger test—ratio, comparison, integral, alternating series, etc. For the Topic 10.3 study guide see (https://library.fiveable.me/ap-calculus/unit-10/nth-term-test-for-divergence/study-guide/oEMEbEp7gWXCgxDFyRlx). For more unit review and practice questions go to (https://library.fiveable.me/ap-calculus/unit-10) and (https://library.fiveable.me/practice/ap-calculus).

Find the limit of a_n as n → ∞ so you can apply the nth-term (divergence) test: if lim a_n ≠ 0 (or doesn’t exist), the series ∑a_n must diverge. If lim a_n = 0, the test is inconclusive. How to compute lim a_n (quick checklist): - Simplify algebraically (divide numerator & denominator by highest power of n). Example: a_n = n/(n+1) → divide by n → 1/(1+1/n) → limit 1 → series diverges. - Compare growth rates: polynomials vs exponentials. n^p / a^n → 0 if |a|>1. So a_n = 2^n/n! → 0. - Recognize standard limits: 1/n^p → 0 (p>0); r^n → 0 iff |r|<1; (−1)^n oscillates (no limit). - Use L’Hôpital on sequences written as functions f(n) → f(x) then x→∞ when appropriate. - Squeeze theorem if you can bound a_n between two sequences with same limit. Examples: a_n = 1/n → 0 (inconclusive); a_n = (−1)^n → no limit → series diverges by nth-term test; a_n = (3/4)^n → 0 (could converge). For AP BC review of the test and worked examples, see the Topic 10.3 study guide (https://library.fiveable.me/ap-calculus/unit-10/nth-term-test-for-divergence/study-guide/oEMEbEp7gWXCgxDFyRlx). Practice lots of limits on Fiveable’s practice page (https://library.fiveable.me/practice/ap-calculus)—knowing common patterns will save you time on the exam.

The nth-term test is just a “zero test”: if lim_{n→∞} a_n ≠ 0 (or doesn’t exist), the series ∑a_n cannot converge—because the terms don’t shrink to 0, the partial sums keep changing by a nonzero amount, so the series diverges. That’s why the test proves divergence. But lim_{n→∞} a_n = 0 is only a necessary condition, not sufficient. Even when terms go to zero, partial sums can still blow up—e.g., the harmonic series ∑1/n has a_n → 0 but diverges. So the nth-term test can only give a one-way conclusion: nonzero limit ⇒ divergence; zero limit ⇒ inconclusive. For deciding convergence you must use other tests (comparison, integral, ratio, alternating, p-series, etc.). This is exactly what the CED emphasizes for LIM-7.A (nth-term test = divergence test). See the Topic 10.3 study guide for examples and practice (https://library.fiveable.me/ap-calculus/unit-10/nth-term-test-for-divergence/study-guide/oEMEbEp7gWXCgxDFyRlx). For more practice, check the unit overview (https://library.fiveable.me/ap-calculus/unit-10) or the practice problems (https://library.fiveable.me/practice/ap-calculus).

If lim_{n→∞} a_n ≠ 0, the series Σ a_n must diverge. This is the nth-term (or divergence/zero) test: a necessary condition for convergence is that the terms go to 0, so if they don’t, the sequence of partial sums cannot settle to a finite limit. Notes that help for AP BC: this test only shows divergence—it can’t prove convergence when lim a_n = 0 (you’ll need other tests). Use the nth-term test early when terms don’t approach 0 (that immediately rules the series out). This is exactly what LIM-7.A.5 in the CED describes. For a refresher, see the Topic 10.3 study guide (https://library.fiveable.me/ap-calculus/unit-10/nth-term-test-for-divergence/study-guide/oEMEbEp7gWXCgxDFyRlx) and more unit review (https://library.fiveable.me/ap-calculus/unit-10). For extra practice, try problems at (https://library.fiveable.me/practice/ap-calculus).

Good question—the nth-term test only tells you when a series definitely diverges, not when it converges. - The test: if lim_{n→∞} a_n ≠ 0 or does not exist, then ∑ a_n diverges. That’s a necessary condition for convergence (zero test). - It’s inconclusive when lim_{n→∞} a_n = 0. Why? Because a_n → 0 is required for convergence but not enough by itself. Examples: - ∑ 1/n (harmonic) has terms → 0 but diverges. - ∑ 1/2^n has terms → 0 and converges (geometric). - Alternating harmonic ∑ (-1)^{n+1}/n has terms → 0 and converges conditionally. So: if the limit isn’t zero, you’re done—divergent. If the limit is zero, you must use other tests (comparison, integral, ratio, alternating series, p-series, etc.) to decide. This is exactly what Topic 10.3 in the CED emphasizes—nth-term test = divergence test, not a convergence test. For the AP review and examples, check the Topic 10.3 study guide (https://library.fiveable.me/ap-calculus/unit-10/nth-term-test-for-divergence/study-guide/oEMEbEp7gWXCgxDFyRlx) and more unit resources (https://library.fiveable.me/ap-calculus/unit-10). For extra practice, try problems at (https://library.fiveable.me/practice/ap-calculus).

If lim_{n→∞} a_n does not exist, the nth-term test (zero test) tells you the series ∑ a_n diverges. The CED’s key idea is that a necessary condition for convergence is that the terms go to 0; if the limit fails to exist (or equals any nonzero number), the series cannot converge. Why: convergence of ∑ a_n implies the sequence of terms a_n → 0 (otherwise partial sums can’t settle). So nonexistence of the limit rules out convergence. Note this is only a divergence test—if lim a_n = 0, the test is inconclusive (you’ll need another test like integral, comparison, ratio, or alternating series). For a quick review, check the Topic 10.3 study guide (https://library.fiveable.me/ap-calculus/unit-10/nth-term-test-for-divergence/study-guide/oEMEbEp7gWXCgxDFyRlx) and more unit resources (https://library.fiveable.me/ap-calculus/unit-10). For extra practice, use the AP problems at (https://library.fiveable.me/practice/ap-calculus).

Start with the nth-term (zero) test every time: if lim a_n ≠ 0, the series diverges immediately (fast check). If lim a_n = 0, that just means “maybe”—you need other tests. Quick strategy for what to try next (order of checks you’ll usually follow on the AP BC exam): - Recognize the form: is it geometric? If so, use the geometric-series test. - Look for p-series or something like 1/n^p → use p-series. - Alternating signs with decreasing terms → try the Alternating Series Test (for conditional convergence). - Positive-term rational combos of polynomials → try Limit Comparison with a known p-series. - Exponentials or factorials → try Ratio or Root Test. - Terms coming from an integral-friendly function (decreasing, positive) → Integral Test. - Obvious telescoping sum → compute partial sums. - If you want absolute vs conditional: test absolute convergence with one of the positive-term tests above. Always state why a test applies (conditions from the CED: e.g., monotone decreasing for the integral or alternating tests). For guided practice and examples on Topic 10.3, see the Fiveable study guide (https://library.fiveable.me/ap-calculus/unit-10/nth-term-test-for-divergence/study-guide/oEMEbEp7gWXCgxDFyRlx). For lots of practice problems, use the AP practice bank (https://library.fiveable.me/practice/ap-calculus).

Short answer: the nth-term (or zero) test applies to any infinite series ∑ a_n, but it only tests for divergence—not convergence. If lim_{n→∞} a_n ≠ 0 or doesn’t exist, the series MUST diverge. If lim_{n→∞} a_n = 0, the test is inconclusive: the series might converge (e.g., alternating harmonic) or diverge (e.g., harmonic series). So restrictions to remember: - You may apply it to any series (geometric, p-series, alternating, etc.) to try to prove divergence. - You cannot use it to prove convergence. If the terms go to 0, you must use other tests (ratio, integral, comparison, alternating series, etc.) per the CED LIM-7.A objectives. For a quick refresher, see the Topic 10.3 study guide (https://library.fiveable.me/ap-calculus/unit-10/nth-term-test-for-divergence/study-guide/oEMEbEp7gWXCgxDFyRlx) and try practice problems at (https://library.fiveable.me/practice/ap-calculus).

Short answer: a sequence is the list of terms a_n; a series is the sum of those terms (partial sums S_n = a_1 + a_2 + ... + a_n). The nth-term test (zero test) looks at the sequence a_n, not the partial-sum sequence S_n. How to use it (CED language): compute lim_{n→∞} a_n. - If lim a_n ≠ 0 (or doesn’t exist), the series ∑ a_n diverges—nonzero term implies divergence (necessary condition). - If lim a_n = 0, the test is inconclusive: the series might converge (e.g., alternating series) or diverge (e.g., harmonic series ∑ 1/n). Quick examples: a_n = 1/n → lim a_n = 0 but ∑ 1/n diverges. a_n = (–1)^n/n → lim = 0 and ∑ converges conditionally. This is a BC-only divergence test (LIM-7.A.5). For a concise guide and practice, see the Topic 10.3 study guide (https://library.fiveable.me/ap-calculus/unit-10/nth-term-test-for-divergence/study-guide/oEMEbEp7gWXCgxDFyRlx) and more unit review (https://library.fiveable.me/ap-calculus/unit-10). For extra practice questions, try Fiveable’s problem bank (https://library.fiveable.me/practice/ap-calculus).