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AP Calculus BC Flashcards: The Nth Term Test For Divergence

Study The Nth Term Test For Divergence in AP Calculus BC with focused flashcards that help you recognize the idea, recall the key rule, and apply it in practice-style prompts.

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What this deck covers

This deck focuses on The Nth Term Test For Divergence, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus BC.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

AP Calculus BC Flashcards: The Nth Term Test For Divergence

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QUESTION

Evaluate lim⁡n→∞3nn+3\lim_{{n \to \infty}} \frac{3n}{n+3}limn→∞​n+33n​. Divergent?

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ANSWER

Limit is 3; series diverges. Leading terms give ratio 31=3≠0\frac{3}{1} = 3 \neq 013​=3=0.

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Flashcard 1: Evaluate lim⁡n→∞3nn+3\lim_{{n \to \infty}} \frac{3n}{n+3}limn→∞​n+33n​. Divergent?

Answer: Limit is 3; series diverges. Leading terms give ratio 31=3≠0\frac{3}{1} = 3 \neq 013​=3=0.

Flashcard 2: Evaluate lim⁡n→∞2n\lim_{{n \to \infty}} \frac{2}{n}limn→∞​n2​. Divergent?

Answer: Limit is 0; test inconclusive. Constant multiple of 1n\frac{1}{n}n1​ still gives limit 0.

Flashcard 3: Assess lim⁡n→∞2n+1n\lim_{{n \to \infty}} \frac{2n+1}{n}limn→∞​n2n+1​. Divergent?

Answer: Limit is 2; series diverges. Dividing gives 2+1n2 + \frac{1}{n}2+n1​, limit 2.

Flashcard 4: What if lim⁡n→∞an≠0\lim_{{n \to \infty}} a_n \neq 0limn→∞​an​=0?

Answer: Series ∑an\sum a_n∑an​ diverges. Non-zero limit always implies divergence by the test.

Flashcard 5: Find lim⁡n→∞n2n3+1\lim_{{n \to \infty}} \frac{n^2}{n^3 + 1}limn→∞​n3+1n2​.

Answer: Limit is 0; test inconclusive. Highest degree terms show limit is 0, test inconclusive.

Flashcard 6: Determine divergence: lim⁡n→∞(−1)n\lim_{n \to \infty} (-1)^nlimn→∞​(−1)n.

Answer: Does not exist; series diverges. Oscillating sequence has no limit, so series diverges.

Flashcard 7: Evaluate lim⁡n→∞1n3\lim_{{n \to \infty}} \frac{1}{n^3}limn→∞​n31​. Divergent?

Answer: Limit is 0; test inconclusive. Higher power in denominator gives limit 0.

Flashcard 8: Assess divergence: lim⁡n→∞n+1n+2\lim_{n \to \infty} \frac{n+1}{n+2}limn→∞​n+2n+1​.

Answer: Limit is 111; series diverges. Both numerator and denominator approach ∞\infty∞, limit is 111.

Flashcard 9: Evaluate lim⁡n→∞nn+1\lim_{n \to \infty} \frac{n}{n+1}limn→∞​n+1n​ for divergence.

Answer: Limit is 111; series diverges. Since limit equals 1≠01 \neq 01=0, the test proves divergence.

Flashcard 10: Evaluate divergence: lim⁡n→∞5n3n\lim_{n \to \infty} \frac{5n}{3n}limn→∞​3n5n​

Answer: Limit is 53\frac{5}{3}35​; series diverges. Coefficients cancel to give limit 53≠0\frac{5}{3} \neq 035​=0

Flashcard 11: What is the nth term test for divergence?

Answer: If lim⁡n→∞an≠0\lim_{n \to \infty} a_n \neq 0limn→∞​an​=0, then ∑an\sum a_n∑an​ diverges. The fundamental condition for proving a series diverges.

Flashcard 12: What does lim⁡n→∞an=L≠0\lim_{{n \to \infty}} a_n = L \neq 0limn→∞​an​=L=0 imply?

Answer: Series ∑an\sum a_n∑an​ diverges. Any non-zero limit guarantees series divergence.

Flashcard 13: Does lim⁡n→∞an=0\lim_{{n \to \infty}} a_n = 0limn→∞​an​=0 guarantee convergence?

Answer: No, it does not guarantee convergence. Zero limit is necessary but not sufficient for convergence.

Flashcard 14: What conclusion if lim⁡n→∞an=0\lim_{{n \to \infty}} a_n = 0limn→∞​an​=0?

Answer: Test is inconclusive. Zero limit cannot determine convergence or divergence.

Flashcard 15: Does lim⁡n→∞an=d≠0\lim_{{n \to \infty}} a_n = d \neq 0limn→∞​an​=d=0 infer divergence?

Answer: Yes, series ∑an\sum a_n∑an​ diverges. Any non-zero constant limit proves divergence.

Flashcard 16: Determine lim⁡n→∞n+5n\lim_{{n \to \infty}} \frac{n + 5}{n}limn→∞​nn+5​. Divergent?

Answer: Limit is 1; series diverges. Adding constant to numerator gives limit 1.

Flashcard 17: Determine divergence: lim⁡n→∞n\lim_{{n \to \infty}} \sqrt{n}limn→∞​n​.

Answer: Limit is ∞\infty∞; series diverges. Square root grows without bound to infinity.

Flashcard 18: What does lim⁡n→∞an=0\lim_{{n \to \infty}} a_n = 0limn→∞​an​=0 indicate?

Answer: Test is inconclusive. Zero limit cannot prove convergence or divergence alone.

Flashcard 19: Identify lim⁡n→∞4n32n3\lim_{{n \to \infty}} \frac{4n^3}{2n^3}limn→∞​2n34n3​. Divergent?

Answer: Limit is 2; series diverges. Coefficients simplify to 42=2≠0\frac{4}{2} = 2 \neq 024​=2=0.

Flashcard 20: Evaluate divergence: an=sin⁡(n)a_n = \sin(n)an​=sin(n).

Answer: lim⁡n→∞an\lim_{{n \to \infty}} a_nlimn→∞​an​ does not exist; series diverges. Sine function oscillates, so limit doesn't exist.

Flashcard 21: Determine divergence: lim⁡n→∞n2+2nn\lim_{{n \to \infty}} \frac{n^2 + 2n}{n}limn→∞​nn2+2n​.

Answer: Limit is ∞\infty∞; series diverges. Simplifies to n+2n + 2n+2, which grows to infinity.

Flashcard 22: What must be true for the nth term test to be applicable?

Answer: ana_nan​ must be the general term of a series. The test applies only to terms of infinite series.

Flashcard 23: Evaluate lim⁡n→∞n2+1n2\lim_{{n \to \infty}} \frac{n^2 + 1}{n^2}limn→∞​n2n2+1​. Divergent?

Answer: Limit is 1; series diverges. The constant term 1 makes limit equal to 1.

Flashcard 24: Determine lim⁡n→∞2nn+1\lim_{{n \to \infty}} \frac{2n}{n+1}limn→∞​n+12n​. Does it diverge?

Answer: Limit is 222; series diverges. Leading coefficients give limit 2≠02 \neq 02=0, proving divergence.

Flashcard 25: Determine divergence: lim⁡n→∞nn2\lim_{{n \to \infty}} \frac{n}{n^2}limn→∞​n2n​.

Answer: Limit is 0; test inconclusive. Denominator grows faster, giving limit 0.

Flashcard 26: Evaluate lim⁡n→∞n3n2+1\lim_{{n \to \infty}} \frac{n^3}{n^2 + 1}limn→∞​n2+1n3​. Divergent?

Answer: Limit is ∞\infty∞; series diverges. Degree of numerator exceeds denominator, so limit is infinite.

Flashcard 27: Check divergence: lim⁡n→∞e−n\lim_{{n \to \infty}} e^{-n}limn→∞​e−n.

Answer: Limit is 0; test inconclusive. Exponential decay gives limit 0.

Flashcard 28: Find lim⁡n→∞1n\lim_{{n \to \infty}} \frac{1}{n}limn→∞​n1​ for the nth term test.

Answer: Limit is 0; test inconclusive. This gives the harmonic series limit of 0.

Flashcard 29: Evaluate lim⁡n→∞1n2\lim_{{n \to \infty}} \frac{1}{n^2}limn→∞​n21​ for divergence.

Answer: Limit is 0; test inconclusive. Zero limit means we need other tests to determine behavior.

Flashcard 30: What does lim⁡n→∞an=0\lim_{{n \to \infty}} a_n = 0limn→∞​an​=0 suggest?

Answer: Test is inconclusive. Zero limit means the test provides no information.