Alternating Series Error Bound - AP Calculus BC

1. What is the maximum possible error for an alternating series approximation?

   Error $\text{E}_n = |a_{n+1}|$. The error equals the absolute value of the next unused term.

2. Find the error bound for the 3rd partial sum of $\sum (-1)^n/n^5$.

   Error is less than or equal to $\frac{1}{4^5} = \frac{1}{1024}$. The 4th term is $\frac{1}{4^5}$ for the 3rd partial sum.

3. Determine if $\sum (-1)^n/(n^4 + n)$ converges.

   Yes, it converges since terms decrease and limit to zero. $\frac{1}{n^4+n}$ decreases and approaches 0.

4. Why must $a_n$ decrease in an alternating series?

   To ensure convergence and a valid error bound. Decreasing terms ensure the error bound formula works.

5. Estimate the error bound for the 5th partial sum of $\sum (-1)^n/n^4$.

   Error is less than or equal to $\frac{1}{6^4} = \frac{1}{1296}$. The 6th term is $\frac{1}{6^4}$ for the 5th partial sum.

6. What does a smaller error bound indicate?

   Greater accuracy of the partial sum. Smaller error means the approximation is more precise.

7. Why is the error bound dependent on $|a_{n+1}|$?

   It measures the maximum error using the next term's size. The next term represents the maximum possible error magnitude.

8. Estimate the error bound for the 6th partial sum of $\sum (-1)^n/n^6$.

   Error is less than or equal to $\frac{1}{7^6}$. The 7th term is $\frac{1}{7^6}$ for the 6th partial sum.

9. Explain the significance of $a_n \to 0$ in the Alternating Series Test.

   It ensures terms decrease to zero, aiding convergence. Without this limit, the series cannot converge.

10. Determine if the series $\sum (-1)^n/n$ satisfies the test conditions.

    Yes, it converges since terms decrease and limit to zero. $\frac{1}{n}$ decreases and approaches 0 as $n \to \infty$.

11. What ensures the accuracy of an alternating series sum?

    A decreasing error bound ensures accuracy. Small error bounds mean the partial sum is close to the true sum.

12. Why does the series $\sum (-1)^n/n^3$ converge?

    Terms decrease and limit to zero. $\frac{1}{n^3}$ decreases and approaches 0 as $n$ increases.

13. Identify the condition for an alternating series to converge.

    Terms must decrease in absolute value and limit to zero. These are the two conditions for the Alternating Series Test.

14. What condition must $a_n$ satisfy for the Alternating Series Test?

    $a_n$ must decrease and limit to zero. Both decreasing and limiting to zero are required.

15. State the conclusion of the Alternating Series Test.

    The series converges if conditions are met. If both conditions hold, the alternating series converges.

16. Predict the convergence of $\sum (-1)^n/(2^n)$.

    Converges, terms decrease and limit to zero. $\frac{1}{2^n}$ decreases geometrically to 0.

17. What is the Alternating Series Test?

    A series converges if terms decrease in absolute value and limit to zero. This is Leibniz's test for alternating series convergence.

18. What role does $(-1)^n$ play in an alternating series?

    It alternates the sign of the terms. It creates the alternating pattern of positive and negative terms.

19. What ensures the validity of an error bound in an alternating series?

    Decreasing terms and $a_n \to 0$ ensure validity. These conditions guarantee the error bound theorem applies.

20. How does the error bound relate to convergence?

    A decreasing error bound indicates convergence. Smaller errors indicate the partial sums approach the true sum.

21. What mathematical concept ensures the error bound holds?

    The non-increasing nature of $a_n$ and its limit to zero. These properties guarantee the error bound formula holds.

22. How is the error bound expressed in terms of $n$?

    Error $E_n \text{ is less than or equal to } |a_{n+1}|$. The error is bounded by the absolute value of the next term.

23. What happens if $a_n$ does not decrease?

    The series may not converge; error bound invalid. Without decreasing terms, the alternating series test fails.

24. What is the importance of the error bound in an alternating series?

    It estimates the maximum error in the sum approximation. It provides a bound on how close the partial sum is to the true sum.

25. Find the error bound for the 4th partial sum of $\sum (-1)^n/n^3$.

    Error is less than or equal to $\frac{1}{5^3} = \frac{1}{125}$. The 5th term gives the error bound for the 4th partial sum.

26. Evaluate the convergence of $\sum (-1)^n/(n^3 + 1)$.

    Converges, terms decrease and limit to zero. $\frac{1}{n^3+1}$ decreases and approaches 0.

27. What is the effect of a larger $n$ on the error bound?

    A larger $n$ decreases the error bound. Higher $n$ means the next term $|a_{n+1}|$ is smaller.

28. What does $|a_{n+1}|$ represent in the error bound formula?

    It represents the absolute value of the next term. This is the magnitude of the first omitted term in the sum.

29. Find the next term error bound for $\text{E}_3$ in $\sum (-1)^n/n^2$.

    Error $\text{E}_3 \text{ is less than or equal to } \frac{1}{16}$. The fourth term is $\frac{1}{4^2} = \frac{1}{16}$.

30. Evaluate if $\sum (-1)^n/n!$ satisfies the Alternating Series Test.

    Yes, terms decrease and limit to zero; it converges. $\frac{1}{n!}$ decreases rapidly and approaches 0.

31. Can the error bound be used for any series?

    No, only for convergent alternating series. The series must be alternating and satisfy the test conditions.

32. Calculate the error bound for $\sum (-1)^n/(n^2 + 1)$ at $n=4$.

    Error is less than or equal to $\frac{1}{5^2 + 1} = \frac{1}{26}$. The 5th term is $\frac{1}{5^2+1}$ for the 4th partial sum.

33. Determine if $\sum (-1)^n/(n+1)$ satisfies the conditions of the test.

    Yes, terms decrease and limit to zero; it converges. $\frac{1}{n+1}$ decreases and approaches 0.

34. What does convergence of an alternating series imply?

    The sum approaches a finite value. The infinite sum equals a specific finite number.

35. What happens to the error as more terms are added?

    The error generally decreases. More terms provide better approximations with smaller errors.

36. What is the role of the next term in the error bound?

    The next term's absolute value sets the error limit. It provides the upper bound for the approximation error.

37. What is the formula for the Alternating Series Error Bound?

    Error $\text{E}_n \text{ is less than or equal to } |a_{n+1}|$. The error is bounded by the absolute value of the next term.

38. Does the series $\sum (-1)^n n$ meet the Alternating Series Test?

    No, terms do not decrease to zero. The terms $n$ increase without bound, violating the test.

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