AP Calculus BC Flashcards: Alternating Series Error Bound

What this deck covers

This deck focuses on Alternating Series Error Bound, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus BC.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

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

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

Answer: 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.

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

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

Flashcard 4: Why must $a_n$ decrease in an alternating series?

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

Flashcard 5: What happens to the error as more terms are added?

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

Flashcard 6: How does the error bound relate to convergence?

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

Flashcard 7: Estimate the error bound for the 5th partial sum of $\sum (-1)^n/n^4$.

Answer: 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.

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

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

Flashcard 9: What does a smaller error bound indicate?

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

Flashcard 10: Why is the error bound dependent on $|a_{n+1}|$?

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

Flashcard 11: Does the series $\sum (-1)^n n$ meet the Alternating Series Test?

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

Flashcard 12: Estimate the error bound for the 6th partial sum of $\sum (-1)^n/n^6$.

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

Flashcard 13: What role does $(-1)^n$ play in an alternating series?

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

Flashcard 14: How is the error bound expressed in terms of $n$?

Answer: 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.

Flashcard 15: Evaluate if $\sum (-1)^n/n!$ satisfies the Alternating Series Test.

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

Flashcard 16: Explain the significance of $a_n \to 0$ in the Alternating Series Test.

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

Flashcard 17: Find the error bound for the 3rd partial sum of $\sum (-1)^n/n^5$.

Answer: 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.

Flashcard 18: Determine if the series $\sum (-1)^n/n$ satisfies the test conditions.

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

Flashcard 19: What ensures the accuracy of an alternating series sum?

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

Flashcard 20: Why does the series $\sum (-1)^n/n^3$ converge?

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

Flashcard 21: Identify the condition for an alternating series to converge.

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

Flashcard 22: What condition must $a_n$ satisfy for the Alternating Series Test?

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

Flashcard 23: Why must $a_n$ decrease in an alternating series?

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

Flashcard 24: State the conclusion of the Alternating Series Test.

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

Flashcard 25: What does $|a_{n+1}|$ represent in the error bound formula?

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

Flashcard 26: Predict the convergence of $\sum (-1)^n/(2^n)$.

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

Flashcard 27: Determine if $\sum (-1)^n/(n^4 + n)$ converges.

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

Flashcard 28: What is the Alternating Series Test?

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

Flashcard 29: What role does $(-1)^n$ play in an alternating series?

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

Flashcard 30: Find the error bound for the 4th partial sum of $\sum (-1)^n/n^3$.

Answer: 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.

Flashcard 31: What ensures the validity of an error bound in an alternating series?

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

Flashcard 32: How does the error bound relate to convergence?

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

Flashcard 33: What mathematical concept ensures the error bound holds?

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

Flashcard 34: What does convergence of an alternating series imply?

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

Flashcard 35: What mathematical concept ensures the error bound holds?

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

Flashcard 36: Identify the condition for an alternating series to converge.

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

Flashcard 37: What condition must $a_n$ satisfy for the Alternating Series Test?

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

Flashcard 38: What ensures the validity of an error bound in an alternating series?

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

Flashcard 39: How is the error bound expressed in terms of $n$?

Answer: 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.

Flashcard 40: Explain the significance of $a_n \to 0$ in the Alternating Series Test.

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

Flashcard 41: What is the role of the next term in the error bound?

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

Flashcard 42: What happens if $a_n$ does not decrease?

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

Flashcard 43: Determine if $\sum (-1)^n/(n+1)$ satisfies the conditions of the test.

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

Flashcard 44: What is the effect of a larger $n$ on the error bound?

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

Flashcard 45: What is the Alternating Series Test?

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

Flashcard 46: What is the importance of the error bound in an alternating series?

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

Flashcard 47: Find the error bound for the 4th partial sum of $\sum (-1)^n/n^3$.

Answer: 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.

Flashcard 48: Evaluate the convergence of $\sum (-1)^n/(n^3 + 1)$.

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

Flashcard 49: State the conclusion of the Alternating Series Test.

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

Flashcard 50: What happens if $a_n$ does not decrease?

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

Flashcard 51: What does a smaller error bound indicate?

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

Flashcard 52: Evaluate the convergence of $\sum (-1)^n/(n^3 + 1)$.

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

Flashcard 53: Why does the series $\sum (-1)^n/n^3$ converge?

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

Flashcard 54: Can the error bound be used for any series?

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

Flashcard 55: What is the effect of a larger $n$ on the error bound?

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

Flashcard 56: Estimate the error bound for the 5th partial sum of $\sum (-1)^n/n^4$.

Answer: 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.

Flashcard 57: What does $|a_{n+1}|$ represent in the error bound formula?

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

Flashcard 58: Why is the error bound dependent on $|a_{n+1}|$?

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

Flashcard 59: Calculate the error bound for $\sum (-1)^n/(n^2 + 1)$ at $n=4$.

Answer: 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.

Flashcard 60: Determine if the series $\sum (-1)^n/n$ satisfies the test conditions.

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

Flashcard 61: Find the next term error bound for $\text{E}_3$ in $\sum (-1)^n/n^2$.

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

Flashcard 62: What is the formula for the Alternating Series Error Bound?

Answer: 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.

Flashcard 63: Evaluate if $\sum (-1)^n/n!$ satisfies the Alternating Series Test.

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

Flashcard 64: Can the error bound be used for any series?

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

Flashcard 65: What is the importance of the error bound in an alternating series?

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

Flashcard 66: Calculate the error bound for $\sum (-1)^n/(n^2 + 1)$ at $n=4$.

Answer: 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.

Flashcard 67: Determine if $\sum (-1)^n/(n+1)$ satisfies the conditions of the test.

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

Flashcard 68: What is the maximum possible error for an alternating series approximation?

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

Flashcard 69: What ensures the accuracy of an alternating series sum?

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

Flashcard 70: Find the next term error bound for $\text{E}_3$ in $\sum (-1)^n/n^2$.

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

Flashcard 71: What does convergence of an alternating series imply?

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

Flashcard 72: What happens to the error as more terms are added?

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

Flashcard 73: What is the role of the next term in the error bound?

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

Flashcard 74: What is the formula for the Alternating Series Error Bound?

Answer: 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.

Flashcard 75: Does the series $\sum (-1)^n n$ meet the Alternating Series Test?

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