AP Calculus BC Flashcards: The Nth Term Test For Divergence

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.

Flashcard 1: Evaluate $\lim_{{n \to \infty}} \frac{n^3}{n^2 + 1}$. Divergent?

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

Flashcard 2: What does $\lim_{{n \to \infty}} a_n = 0$ suggest?

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

Flashcard 3: What does $\lim_{{n \to \infty}} a_n = 0$ indicate?

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

Flashcard 4: Evaluate $\lim_{{n \to \infty}} \frac{2}{n}$. Divergent?

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

Flashcard 5: Identify $\lim_{{n \to \infty}} \frac{4n^3}{2n^3}$. Divergent?

Answer: Limit is 2; series diverges. Coefficients simplify to $\frac{4}{2} = 2 \neq 0$.

Flashcard 6: What must be true for the nth term test to be applicable?

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

Flashcard 7: Determine $\lim_{{n \to \infty}} \frac{n + 5}{n}$. Divergent?

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

Flashcard 8: Evaluate $\lim_{{n \to \infty}} \frac{3n}{n+3}$. Divergent?

Answer: Limit is 3; series diverges. Leading terms give ratio $\frac{3}{1} = 3 \neq 0$.

Flashcard 9: Does $\lim_{{n \to \infty}} a_n = d \neq 0$ infer divergence?

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

Flashcard 10: Determine divergence: $\lim_{{n \to \infty}} \frac{n}{n^2}$.

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

Flashcard 11: What is the outcome if $\lim_{{n \to \infty}} a_n = c \neq 0$?

Answer: Series $\sum a_n$ diverges. Any constant $c \neq 0$ proves divergence.

Flashcard 12: Determine divergence: $\lim_{{n \to \infty}} \sqrt{n}$.

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

Flashcard 13: Identify divergence: $\lim_{{n \to \infty}} \ln(n)$.

Answer: Limit is $\infty$; series diverges. Natural log grows without bound to infinity.

Flashcard 14: Evaluate $\lim_{{n \to \infty}} \frac{1}{n^3}$. Divergent?

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

Flashcard 15: What does $\lim_{{n \to \infty}} a_n = 0$ suggest?

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

Flashcard 16: Check divergence: $\lim_{{n \to \infty}} e^{-n}$.

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

Flashcard 17: Check divergence: $\lim_{n \to \infty} e^{-n}$.

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

Flashcard 18: Evaluate divergence: $a_n = \sin(n)$.

Answer: $\lim_{{n \to \infty}} a_n$ does not exist; series diverges. Sine function oscillates, so limit doesn't exist.

Flashcard 19: Determine divergence: $\lim_{{n \to \infty}} (-1)^n$.

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

Flashcard 20: Identify $\lim_{{n \to \infty}} \frac{3n^2}{5n^2}$. Divergent?

Answer: Limit is $\frac{3}{5}$; series diverges. Equal highest degree terms give ratio $\frac{3}{5} \neq 0$.

Flashcard 21: What conclusion if $\lim_{{n \to \infty}} a_n = 0$?

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

Flashcard 22: What is the nth term test for divergence?

Answer: If $\lim_{n \to \infty} a_n \neq 0$, then $\sum a_n$ diverges. The fundamental condition for proving a series diverges.

Flashcard 23: Evaluate $\lim_{{n \to \infty}} \frac{n}{n+1}$ for divergence.

Answer: Limit is 1; series diverges. Since limit equals 1 $≠$ 0, the test proves divergence.

Flashcard 24: Determine divergence: $\lim_{{n \to \infty}} \frac{n^2 + 2n}{n}$.

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

Flashcard 25: Determine divergence: $\lim_{n \to \infty} \sqrt{n}$.

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

Flashcard 26: Determine $\lim_{{n \to \infty}} \frac{2n}{n+1}$. Does it diverge?

Answer: Limit is 2; series diverges. Leading coefficients give limit 2 ≠ 0, proving divergence.

Flashcard 27: Evaluate $\lim_{{n \to \infty}} \frac{2}{n}$. Divergent?

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

Flashcard 28: Determine $\lim_{{n \to \infty}} \frac{2n}{n+1}$. Does it diverge?

Answer: Limit is $2$; series diverges. Leading coefficients give limit $2 \neq 0$, proving divergence.

Flashcard 29: Does $\lim_{{n \to \infty}} \frac{n+1}{n}$ imply divergence?

Answer: Limit is 1; series diverges. Adding 1 to numerator gives limit 1 ≠ 0.

Flashcard 30: Find $\lim_{{n \to \infty}} \frac{n^2}{n^3 + 1}$.

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

Flashcard 31: Does $\lim_{{n \to \infty}} a_n = 0$ guarantee convergence?

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

Flashcard 32: What does $\lim_{{n \to \infty}} a_n = L \neq 0$ imply?

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

Flashcard 33: Identify $\lim_{{n \to \infty}} \frac{n^4}{n^3}$. Divergence?

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

Flashcard 34: Identify $\lim_{{n \to \infty}} \frac{3n^2}{5n^2}$. Divergent?

Answer: Limit is $\frac{3}{5}$; series diverges. Equal highest degree terms give ratio $\frac{3}{5} \neq 0$.

Flashcard 35: Evaluate $\lim_{{n \to \infty}} \frac{n^2 + 1}{n^2}$. Divergent?

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

Flashcard 36: Evaluate divergence: $\lim_{n \to \infty} \frac{5n}{3n}$

Answer: Limit is $\frac{5}{3}$; series diverges. Coefficients cancel to give limit $\frac{5}{3} \neq 0$

Flashcard 37: Identify $\lim_{{n \to \infty}} \frac{4n^3}{2n^3}$. Divergent?

Answer: Limit is 2; series diverges. Coefficients simplify to $\frac{4}{2} = 2 \neq 0$.

Flashcard 38: Find $\lim_{{n \to \infty}} \frac{n^2}{n^3 + 1}$.

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

Flashcard 39: Evaluate divergence: $\lim_{n \to \infty} \frac{5n}{3n}$.

Answer: Limit is $\frac{5}{3}$; series diverges. Coefficients cancel to give limit $\frac{5}{3} \neq 0$.

Flashcard 40: Identify divergence: $\lim_{n \to \infty} \ln(n)$.

Answer: Limit is $\infty$; series diverges. Natural log grows without bound to infinity.

Flashcard 41: Evaluate $\lim_{n \to \infty} \frac{n}{n+1}$ for divergence.

Answer: Limit is $1$; series diverges. Since limit equals $1 \neq 0$, the test proves divergence.

Flashcard 42: What does $\lim_{{n \to \infty}} a_n = L \neq 0$ imply?

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

Flashcard 43: What if $\lim_{{n \to \infty}} a_n \neq 0$?

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

Flashcard 44: Evaluate divergence: $a_n = \sin(n)$.

Answer: $\lim_{{n \to \infty}} a_n$ does not exist; series diverges. Sine function oscillates, so limit doesn't exist.

Flashcard 45: Assess $\lim_{{n \to \infty}} \frac{2n+1}{n}$. Divergent?

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

Flashcard 46: Determine divergence: $\lim_{n \to \infty} (-1)^n$.

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

Flashcard 47: Does $\lim_{{n \to \infty}} a_n = d \neq 0$ infer divergence?

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

Flashcard 48: Determine $\lim_{{n \to \infty}} \frac{n + 5}{n}$. Divergent?

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

Flashcard 49: Find $\lim_{n \to \infty} \frac{1}{n}$ for the nth term test.

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

Flashcard 50: When can the nth term test conclude divergence?

Answer: When $\lim_{{n \to \infty}} a_n \neq 0$. Only when the limit is non-zero can we conclude divergence.

Flashcard 51: What conclusion if $\lim_{{n \to \infty}} a_n = 0$?

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

Flashcard 52: What must be true for the nth term test to be applicable?

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

Flashcard 53: Assess divergence: $\lim_{n \to \infty} \frac{n+1}{n+2}$.

Answer: Limit is $1$; series diverges. Both numerator and denominator approach $\infty$, limit is $1$.

Flashcard 54: Does $\lim_{{n \to \infty}} a_n = 0$ guarantee convergence?

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

Flashcard 55: Evaluate $\lim_{{n \to \infty}} \frac{1}{n^2}$ for divergence.

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

Flashcard 56: Does $\lim_{{n \to \infty}} \frac{n+1}{n}$ imply divergence?

Answer: Limit is 1; series diverges. Adding 1 to numerator gives limit 1 ≠ 0.

Flashcard 57: Evaluate $\lim_{{n \to \infty}} \frac{n^2 + 1}{n^2}$. Divergent?

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

Flashcard 58: Evaluate $\lim_{{n \to \infty}} \frac{3n}{n+3}$. Divergent?

Answer: Limit is 3; series diverges. Leading terms give ratio $\frac{3}{1} = 3 \neq 0$.

Flashcard 59: What is the outcome if $\lim_{{n \to \infty}} a_n = c \neq 0$?

Answer: Series $\sum a_n$ diverges. Any constant $c \neq 0$ proves divergence.

Flashcard 60: Identify $\lim_{{n \to \infty}} \frac{n^4}{n^3}$. Divergence?

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

Flashcard 61: What is the nth term test for divergence?

Answer: If $\lim_{n \to \infty} a_n \neq 0$, then $\sum a_n$ diverges. The fundamental condition for proving a series diverges.

Flashcard 62: What role does the nth term test play in series analysis?

Answer: Primarily to identify divergence. It's a quick first test to eliminate divergent series.

Flashcard 63: Evaluate $\lim_{{n \to \infty}} \frac{n^3}{n^2 + 1}$. Divergent?

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

Flashcard 64: What if $\lim_{{n \to \infty}} a_n \neq 0$?

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

Flashcard 65: Assess divergence: $\lim_{{n \to \infty}} \frac{n+1}{n+2}$.

Answer: Limit is 1; series diverges. Both numerator and denominator approach infinity, limit is 1.

Flashcard 66: Find $\lim_{{n \to \infty}} \frac{1}{n}$ for the nth term test.

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

Flashcard 67: What role does the nth term test play in series analysis?

Answer: Primarily to identify divergence. It's a quick first test to eliminate divergent series.

Flashcard 68: What does $\lim_{{n \to \infty}} a_n = 0$ indicate?

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

Flashcard 69: Evaluate $\lim_{{n \to \infty}} \frac{1}{n^2}$ for divergence.

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

Flashcard 70: Determine divergence: $\lim_{{n \to \infty}} \frac{n}{n^2}$.

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

Flashcard 71: Assess $\lim_{n \to \infty} \frac{2n+1}{n}$. Divergent?

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

Flashcard 72: Determine divergence: $\lim_{{n \to \infty}} \frac{n^2 + 2n}{n}$.

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

Flashcard 73: Evaluate $\lim_{{n \to \infty}} \frac{1}{n^3}$. Divergent?

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