The nth Term Test for Divergence - AP Calculus BC

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

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

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

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

3. Evaluate $\lim_{{n \to \infty}} \frac{2}{n}$. Divergent?

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

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

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

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

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

6. Evaluate $\lim_{{n \to \infty}} \frac{3n}{n+3}$. Divergent?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

16. What conclusion if $\lim_{{n \to \infty}} a_n = 0$?

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

17. What is the nth term test for divergence?

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

18. Evaluate $\lim_{{n \to \infty}} \frac{n}{n+1}$ for divergence.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

27. Evaluate $\lim_{n \to \infty} \frac{n}{n+1}$ for divergence.

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

28. What does $\lim_{{n \to \infty}} a_n = L \neq 0$ imply?

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

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

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

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

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

31. Does $\lim_{{n \to \infty}} a_n = d \neq 0$ infer divergence?

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

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

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

33. When can the nth term test conclude divergence?

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

34. What must be true for the nth term test to be applicable?

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

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

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

36. Does $\lim_{{n \to \infty}} a_n = 0$ guarantee convergence?

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

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

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

38. Does $\lim_{{n \to \infty}} \frac{n+1}{n}$ imply divergence?

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

39. What is the outcome if $\lim_{{n \to \infty}} a_n = c \neq 0$?

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

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

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

41. What role does the nth term test play in series analysis?

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

42. What if $\lim_{{n \to \infty}} a_n \neq 0$?

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

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

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

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

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

45. What does $\lim_{{n \to \infty}} a_n = 0$ indicate?

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

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

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

Study these flashcards