Alternating Series Test for Convergence - AP Calculus BC

1. Find if the series $\textstyle \sum_{n=1}^{\infty} (-1)^n \frac{1}{n^5}$ converges.

   Yes, it converges by the Alternating Series Test. Higher powers guarantee all alternating series test conditions are satisfied.

2. What is the Alternating Series Estimation Theorem?

   Provides error bound for partial sums of convergent series. It quantifies how close partial sums are to the series sum.

3. Determine whether $\textstyle \sum_{n=1}^{\infty} (-1)^n n$ converges.

   No, the terms do not approach zero. The terms $a_n = n$ grow without bound, violating the limit condition.

4. Does the series $\textstyle\sum_{n=1}^{\infty} (-1)^n (1 + \frac{1}{n^2})$ converge?

   No, it does not converge. The limit $\lim_{n\to\infty} (1 + \frac{1}{n^2}) = 1 \neq 0$.

5. Find if $\textstyle\bigsum_{n=1}^{\text{inf}} (-1)^n \frac{1}{\text{ln}(n+1)}$ converges.

   Yes, it converges by the Alternating Series Test. All conditions are met: positive terms, decreasing, limit to zero.

6. Does the series $\textstyle\bigsum_{n=1}^{\text{inf}} (-1)^n \frac{1}{n}$ satisfy $a_n \to 0$?

   Yes, $a_n = \frac{1}{n} \to 0$ as $n \to \text{inf}$. The harmonic terms $\frac{1}{n}$ clearly approach zero as $n \to \infty$.

7. What happens if $a_n$ does not decrease to 0 in an alternating series?

   The series diverges. The series fails to meet the necessary convergence conditions.

8. Does the series $\textstyle\bigsum_{n=1}^{\text{inf}} (-1)^n \frac{1}{n^2}$ converge?

   Yes, it converges by the Alternating Series Test. All three conditions are satisfied: positive, decreasing, limit to zero.

9. Determine if $\textstyle\bigsum_{n=1}^{\infty} (-1)^n \frac{1}{n+1}$ satisfies $a_n \to 0$.

   Yes, $a_n = \frac{1}{n+1} \to 0$ as $n \to \infty$. The shifted harmonic terms still approach zero as required.

10. What does it mean if a series is absolutely convergent?

    The series $\textstyle\bigsum |a_n|$ converges. This means the series of absolute values also converges.

11. State the limit condition for the Alternating Series Test.

    $\textstyle\biglim_{n \to \text{inf}} a_n = 0$. The terms must approach zero for the series to have a chance at convergence.

12. Find if the series $\textstyle\sum_{n=1}^{\infty} (-1)^n \frac{1}{n^5}$ converges.

    Yes, it converges by the Alternating Series Test. Higher powers guarantee all alternating series test conditions are satisfied.

13. Find if $\textstyle\sum_{n=1}^{\infty} (-1)^n \frac{1}{\ln(n+1)}$ converges.

    Yes, it converges by the Alternating Series Test. All conditions are met: positive terms, decreasing, limit to zero.

14. Find the error bound for $\textstyle \sum_{n=1}^{\infty} (-1)^n \frac{1}{n^2}$ at $N=3$.

    Error $|R_3| < |a_4| = \frac{1}{16}$. The fourth term gives the error bound: $\frac{1}{4^2} = \frac{1}{16}$.

15. Does the series $\textstyle \sum_{n=1}^{\infty} (-1)^n (1 + \frac{1}{n^2})$ converge?

    No, it does not converge. The limit $\lim_{n\to\infty} (1 + \frac{1}{n^2}) = 1 \neq 0$.

16. What is the significance of the term $(-1)^n$ in an alternating series?

    It causes the series to alternate in sign. The $(-1)^n$ factor creates the required alternating pattern.

17. Find the limit of $a_n$ for $\textstyle\bigsum_{n=1}^{\text{inf}} (-1)^n \frac{1}{n}$.

    Limit $a_n = \frac{1}{n} \to 0$ as $n \to \text{inf}$. As $n$ increases, $\frac{1}{n}$ approaches zero.

18. Find if $\sum_{n=1}^{\infty} (-1)^n \frac{1}{n^4}$ converges.

    Yes, it converges by the Alternating Series Test. Even higher powers guarantee faster convergence to zero.

19. State the Alternating Series Remainder Theorem.

    The remainder is less than the first unused term. This gives an upper bound on the approximation error.

20. Identify the condition required for terms in the Alternating Series.

    The terms $a_n$ must be positive: $a_n > 0$. This ensures the series doesn't have negative terms interfering with convergence.

21. Does the series $\sum_{n=1}^{\infty} (-1)^n \frac{1}{n^2}$ converge?

    Yes, it converges by the Alternating Series Test. All three conditions are satisfied: positive, decreasing, limit to zero.

22. What does it mean for the terms to be 'eventually decreasing'?

    There exists $N$ such that $a_{n+1} < a_n$ for $n > N$. The decreasing condition only needs to hold for sufficiently large $n$.

23. Determine if $\sum_{n=1}^{\infty} (-1)^n \frac{1}{n+1}$ satisfies $a_n \to 0$.

    Yes, $a_n = \frac{1}{n+1} \to 0$ as $n \to \infty$. The shifted harmonic terms still approach zero as required.

24. Determine if the series $\sum_{n=1}^{\infty} (-1)^n \frac{1}{n^3}$ converges.

    Yes, it converges by the Alternating Series Test. Higher powers ensure faster convergence with all conditions satisfied.

25. Determine if the series $\textstyle \sum_{n=1}^{\infty} (-1)^n \frac{1}{n^3}$ converges.

    Yes, it converges by the Alternating Series Test. Higher powers ensure faster convergence with all conditions satisfied.

26. What type of series does the Alternating Series Test apply to?

    Applies to series with terms $(-1)^n a_n$. The alternating factor $(-1)^n$ creates the sign pattern.

27. Identify the series that converges: (A) $\textstyle\bigsum (-1)^n \frac{1}{n^2}$, (B) $\textstyle\bigsum (-1)^n n$.

    (A) converges by the Alternating Series Test. Series (A) satisfies all conditions while (B) has terms that don't approach zero.

28. What is the result if $a_n \to 0$ is not satisfied?

    The series diverges. Without the limit condition, the series cannot converge.

29. State the primary purpose of the Alternating Series Test.

    To determine if an alternating series converges. It checks the three key conditions for alternating series convergence.

30. Identify the series that converges: (A) $\sum (-1)^n \frac{1}{n^2}$, (B) $\sum (-1)^n n$.

    (A) converges by the Alternating Series Test. Series (A) satisfies all conditions while (B) has terms that don't approach zero.

31. Find if $\textstyle \sum_{n=1}^{\infty} (-1)^n \frac{1}{n^4}$ converges.

    Yes, it converges by the Alternating Series Test. Even higher powers guarantee faster convergence to zero.

32. What does the Alternating Series Test determine about a series?

    Determines if an alternating series converges. It's the specific test for alternating series convergence criteria.

33. Find the error bound for $\textstyle\sum_{n=1}^{\infty} (-1)^n \frac{1}{n^2}$ at $N=3$.

    Error $|R_3| < |a_4| = \frac{1}{16}$. The fourth term gives the error bound: $\frac{1}{4^2} = \frac{1}{16}$.

34. Find if the series $\sum_{n=1}^{\infty} (-1)^n \frac{1}{n}$ satisfies decreasing terms.

    Yes, $\frac{1}{n+1} < \frac{1}{n}$ for $n \to \infty$. Since $n+1 > n$, the reciprocals decrease monotonically.

35. What is the error bound for an alternating series?

    Error $|R_N| < |a_{N+1}|$ for partial sum $S_N$. The error magnitude is bounded by the next term's absolute value.

36. Identify if $\textstyle\sum_{n=1}^{\infty} (-1)^n (1 + \frac{1}{n})$ converges.

    No, it does not satisfy $a_n \to 0$ as $n \to \infty$. The limit is $1$, not $0$, so the series diverges.

37. Find the limit of $a_n$ for $\sum_{n=1}^{\infty} (-1)^n \frac{1}{n}$.

    Limit $a_n = \frac{1}{n} \to 0$ as $n \to \infty$. As $n$ increases, $\frac{1}{n}$ approaches zero.

38. Find if the series $\sum_{n=1}^{\infty} (-1)^n \frac{n}{n+1}$ converges.

    No, it does not converge. The limit $\lim_{n\to\infty} \frac{n}{n+1} = 1 \neq 0$.

39. Identify the series that does not converge: (A) $\textstyle\sum (-1)^n \frac{1}{n^3}$, (B) $\textstyle\sum (-1)^n n$.

    (B) does not converge. Series (B) has unbounded terms while (A) satisfies all conditions.

40. Find the error bound for $\textstyle\sum_{n=1}^{\infty} (-1)^n \frac{1}{n}$ at $N=2$.

    Error $|R_2| < |a_3| = \frac{1}{3}$. The third term in the harmonic series: $\frac{1}{3}$.

41. Find the error bound for $\textstyle \sum_{n=1}^{\infty} (-1)^n \frac{1}{n}$ at $N=2$.

    Error $|R_2| < |a_3| = \frac{1}{3}$. The third term in the harmonic series: $\frac{1}{3}$

42. What is the condition for the terms $a_n$ to be considered decreasing?

    $a_{n+1} < a_n$ for all $n$. This ensures the terms form a monotonically decreasing sequence.

43. Find if the series $\textstyle\sum_{n=1}^{\infty} (-1)^n \frac{n}{n+1}$ converges.

    No, it does not converge. The limit $\lim_{n\to\infty} \frac{n}{n+1} = 1 \neq 0$.

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