Determining Absolute or Conditional Convergence - AP Calculus BC

1. What is the first step in the Alternating Series Test?

   Check if $a_n$ is decreasing. Monotonic decreasing requirement.

2. State the condition for a series to converge using the Alternating Series Test.

   $a_n$ decreases to $0$ and $a_n > a_{n+1}$. Terms must decrease monotonically to zero.

3. What is required for the Limit Comparison Test?

   A second series $b_n$ to compare with $a_n$. Need comparison series with known behavior.

4. What test compares series to determine absolute convergence?

   The Comparison Test. Compares term by term with known series.

5. What is the series $\sum_{n=1}^{\infty} \frac{1}{n}$ known as?

   Harmonic Series. Famous divergent series.

6. Apply the Root Test: $\sum_{n=1}^{\infty} \left(\frac{n}{n+1}\right)^n$.

   Diverges. Root approaches 1, so diverges.

7. Identify the series: $\sum_{n=0}^{\infty} x^n$ for $|x| < 1$.

   Geometric Series. Series with constant ratio between terms.

8. What is the condition for convergence using the Root Test?

   If $\lim_{n \to \infty} \sqrt[n]{|a_n|} < 1$, converges absolutely. nth root of terms approaches value less than 1.

9. What is the definition of absolute convergence?

   A series converges absolutely if the series of absolute values converges. Convergence when taking absolute values.

10. What is a necessary condition for series convergence?

    The terms must approach zero as $n \to \infty$. Required for any convergent series.

11. Determine convergence: $\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}$.

    Converges conditionally. Alternating p-series with $p=1/2<1$.

12. Determine convergence: $\sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}$.

    Converges absolutely. Alternating p-series with $p=2>1$.

13. Apply the Root Test: $\sum_{n=1}^{\infty} \left(\frac{1}{3}\right)^n$.

    Converges absolutely. Geometric series with ratio $1/3 < 1$.

14. Explain the role of absolute values in convergence tests.

    Used to test absolute convergence. Determines if series converges absolutely.

15. Determine convergence: $\sum_{n=1}^{\infty} \frac{1}{n}$.

    Diverges. P-series with $p=1$ diverges.

16. Determine convergence: $\sum_{n=1}^{\infty} n^2 e^{-n}$.

    Converges absolutely. Exponential decay dominates polynomial growth.

17. What test determines absolute convergence using absolute values?

    The Absolute Convergence Test. Tests convergence of $\sum |a_n|$.

18. What characterizes a geometric series?

    Each term is a constant multiple of the previous term. Common ratio between consecutive terms.

19. What is the definition of conditional convergence?

    A series converges conditionally if it converges, but not absolutely. Series converges but not when absolute values are taken.

20. Apply the p-series test: $\sum_{n=1}^{\infty} \frac{1}{n^{3/2}}$.

    Converges absolutely. P-series with $p=3/2>1$ converges.

21. Apply the Comparison Test: $\sum_{n=1}^{\infty} \frac{1}{n^2 + 1}$.

    Converges absolutely. Compare with convergent p-series $\sum \frac{1}{n^2}$.

22. What test can determine if a series converges conditionally?

    The Alternating Series Test. For series with alternating signs.

23. What is the p-series test condition for convergence?

    If $p > 1$, the p-series $\sum_{n=1}^{\infty} \frac{1}{n^p}$ converges. Exponent determines convergence behavior.

24. What does the Root Test involve calculating?

    $\lim_{n \to \infty} \sqrt[n]{|a_n|}$. Formula for root test calculation.

25. Determine convergence: $\sum_{n=0}^{\infty} x^n$ for $|x| < 1$.

    Converges absolutely. Common ratio satisfies convergence condition.

26. Determine convergence: $\sum_{n=1}^{\infty} \frac{1}{n^{0.5}}$.

    Diverges. P-series with $p=0.5<1$ diverges.

27. Determine convergence: $\sum_{n=1}^{\infty} (-1)^n \frac{1}{n}$.

    Converges conditionally. Passes Alternating Series Test but absolute values diverge.

28. Identify the limit for convergence in the Ratio Test.

    $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$. Formula for ratio test calculation.

29. What is the Ratio Test's condition for absolute convergence?

    If $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1$, converges absolutely. Ratio of consecutive terms approaches value less than 1.

30. Determine convergence: $\sum_{n=1}^{\infty} \frac{1}{n^2}$.

    Converges absolutely. P-series with $p=2>1$ converges.

31. Which test is most suitable for factorial expressions?

    The Ratio Test. Effective for factorial terms.

32. Identify the series: $\sum_{n=1}^{\infty} (-1)^n \frac{1}{n}$.

    Alternating Harmonic Series. Classic conditionally convergent series.

33. Apply the Ratio Test: $\sum_{n=1}^{\infty} \frac{n!}{(2n)!}$.

    Converges absolutely. Factorials grow faster than exponentials.

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