Ratio Test for Convergence - AP Calculus BC

1. Evaluate $a_n = \frac{n^3}{n!}$ using the Ratio Test. Convergent or divergent?

   Convergent. $L = \frac{(n+1)^3}{n^3} \cdot \frac{1}{n+1} = \frac{(n+1)^2}{n^3} \to 0 < 1$.

2. Is $a_n = \frac{1}{2^n}$ convergent by the Ratio Test?

   Convergent. $L = \frac{1}{2} < 1$.

3. What does the Ratio Test say about $a_n = \frac{1}{(n+1)!}$?

   Convergent. $L = \frac{1}{n+2} \to 0 < 1$.

4. Determine the convergence of $a_n = 3^n \cdot n!$ using the Ratio Test.

   Divergent. $L = 3(n+1) \to \infty > 1$.

5. Evaluate $a_n = \frac{n!}{2^n}$ using the Ratio Test. Convergent or divergent?

   Divergent. $L = \frac{n+1}{2} \to \infty > 1$.

6. Is $a_n = \frac{n!}{n^n}$ convergent by the Ratio Test?

   Convergent. $L = \frac{n+1}{n} \cdot \frac{n^n}{(n+1)^{n+1}} = \frac{1}{e} < 1$.

7. State the limit test result for $a_n = \frac{n^2}{2^n}$ using the Ratio Test.

   Convergent. $L = \frac{(n+1)^2}{n^2} \cdot \frac{1}{2} = \frac{1}{2} < 1$.

8. What does the Ratio Test conclude for $a_n = n^2 e^{-n}$?

   Convergent. $L = \frac{(n+1)^2}{n^2} \cdot \frac{1}{e} = \frac{1}{e} < 1$.

9. Identify the series type: $a_n = \frac{1}{n!}$. Use the Ratio Test.

   Convergent. $L = \frac{1}{(n+1)} \to 0 < 1$.

10. State the Ratio Test conclusion for $a_n = \frac{2^n}{n!}$.

    Convergent. $L = \frac{2}{n+1} \to 0 < 1$.

11. State the formula used in the Ratio Test for a series $a_n$.

    $L = \frac{|a_{n+1}|}{|a_n|}$ as $n \to \infty$. This is the limit of consecutive term ratios.

12. What is the result of Ratio Test for $a_n = \frac{n!}{n^n}$?

    Convergent. $L = \frac{n+1}{n} \cdot \frac{n^n}{(n+1)^{n+1}} = \frac{1}{e} < 1$.

13. Determine the Ratio Test result for $a_n = n^3 e^{-n}$.

    Convergent. $L = \frac{(n+1)^3}{n^3} \cdot \frac{1}{e} = \frac{1}{e} < 1$.

14. Identify the result of the Ratio Test for $a_n = \frac{3^n}{n!}$.

    Convergent. $L = \frac{3}{n+1} \to 0 < 1$.

15. Does $a_n = \frac{n}{2^n}$ converge according to the Ratio Test?

    Convergent. $L = \frac{n+1}{2n} = \frac{1}{2} < 1$.

16. Apply the Ratio Test: $a_n = \frac{n!}{3^n}$. Convergent or divergent?

    Divergent. $L = \frac{n+1}{3} \to \infty > 1$.

17. Does $a_n = \frac{n^2}{(n+1)!}$ converge according to the Ratio Test?

    Convergent. $L = \frac{(n+1)^2}{(n+2)!} \cdot \frac{(n+1)!}{n^2} = \frac{(n+1)^2}{n^2(n+2)} \to 0 < 1$.

18. Identify the convergence of $a_n = \frac{1}{3^n}$ using the Ratio Test.

    Convergent. $L = \frac{1}{3} < 1$.

19. Is $a_n = \frac{5^n}{n!}$ convergent by the Ratio Test?

    Convergent. $L = \frac{5}{n+1} \to 0 < 1$.

20. Apply the Ratio Test: $a_n = \frac{5^n}{n^2}$. Convergent or divergent?

    Divergent. $L = 5 \cdot \frac{n^2}{(n+1)^2} = 5 > 1$.

21. What condition must $L$ satisfy for the series to diverge?

    $L > 1$ or $L = \infty$. When the ratio exceeds 1, terms grow without bound.

22. What is the Ratio Test outcome for $a_n = n!$?

    Divergent. $L = n+1 \to \infty > 1$.

23. Evaluate convergence of $a_n = \frac{2^n}{n!}$ using the Ratio Test.

    Convergent. $L = \frac{2}{n+1} \to 0 < 1$.

24. What does $L = 1$ imply about the convergence of the series?

    The test is inconclusive. The boundary case provides no information about convergence.

25. Does $a_n = \frac{n^2}{3^n}$ converge according to the Ratio Test?

    Convergent. $L = \frac{(n+1)^2}{3n^2} = \frac{1}{3} < 1$.

26. State the Ratio Test result for $a_n = \frac{(n+1)!}{n!}$.

    Divergent. $L = \frac{(n+2)!}{(n+1)!} = n+2 \to \infty > 1$.

27. Apply the Ratio Test: $a_n = \frac{n^2}{n!}$. Convergent or divergent?

    Convergent. $L = \frac{(n+1)^2}{n^2} \cdot \frac{1}{n+1} = \frac{n+1}{n^2} \to 0 < 1$.

28. What condition must $L$ satisfy for the series to converge?

    $L < 1$. When the ratio is less than 1, terms shrink fast enough.

29. What is the convergence outcome for $a_n = \frac{n}{3^n}$ using the Ratio Test?

    Convergent. $L = \frac{n+1}{3n} = \frac{1}{3} < 1$.

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