AP Calculus BC Flashcards: Defining Convergent And Divergent Infinite Series

What this deck covers

This deck focuses on Defining Convergent And Divergent Infinite Series, 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 Limit Comparison Test?

Answer: A test comparing limits of two series to determine convergence. Compares behavior of two series at infinity.

Flashcard 2: Identify whether the series $1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \text{...}$ is convergent.

Answer: Convergent. It is an alternating series. Terms decrease and approach zero, satisfying AST.

Flashcard 3: What is the $n$-th partial sum of the series $1 + \frac{1}{2} + \frac{1}{3} + \text{...}$?

Answer: $S_n = 1 + \frac{1}{2} + \frac{1}{3} + \text{...} + \frac{1}{n}$. Sum of first $n$ terms of the harmonic series.

Flashcard 4: What test would you use for the series $\frac{1}{n^2+n}$?

Answer: Comparison Test or Limit Comparison Test. Best suited for rational function series.

Flashcard 5: When does the Root Test conclude convergence?

Answer: If $\lim_{n \to \infty} \sqrt[n]{|a_n|} < 1$. Limit less than 1 indicates series convergence.

Flashcard 6: What is a telescoping series?

Answer: A series where intermediate terms cancel out. Partial fractions create canceling patterns.

Flashcard 7: What is the Alternating Series Test?

Answer: A test to determine convergence of series with alternating signs. Applies to series of form $\sum (-1)^n a_n$.

Flashcard 8: State the condition for a geometric series to be convergent.

Answer: The common ratio $|r| < 1$. This ensures each term becomes smaller than the previous.

Flashcard 9: When is a series conditionally convergent?

Answer: When it converges, but its absolute value does not. Converges but not absolutely convergent.

Flashcard 10: What is the harmonic series?

Answer: The series $1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \text{...}$. Famous divergent series despite terms approaching zero.

Flashcard 11: Identify the convergence of $\sum (-1)^n \frac{1}{n}$.

Answer: Convergent. It is an alternating harmonic series. Alternating signs with decreasing terms ensure convergence.

Flashcard 12: When does the Ratio Test conclude convergence?

Answer: If $\lim_{n \to \infty} | \frac{a_{n+1}}{a_n} | < 1$. Limit less than 1 guarantees series convergence.

Flashcard 13: State the condition for a geometric series to be convergent.

Answer: The common ratio $|r| < 1$. This ensures each term becomes smaller than the previous.

Flashcard 14: Identify the convergence of $\sum (-1)^n \frac{1}{n}$.

Answer: Convergent. It is an alternating harmonic series. Alternating signs with decreasing terms ensure convergence.

Flashcard 15: Determine if the series $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \text{...}$ converges.

Answer: Convergent. It is a geometric series with $r = \frac{1}{2}$. Since $|\frac{1}{2}| < 1$, the series converges.

Flashcard 16: What is the P-Series Test?

Answer: A test used on series of form $\frac{1}{n^p}$ to determine convergence. Determines convergence for power series forms.

Flashcard 17: Identify if the series $(-1)^n \frac{1}{\text{ln}(n+1)}$ converges.

Answer: Convergent. It satisfies the Alternating Series Test. Terms decrease in magnitude and approach zero.

Flashcard 18: What is an absolutely convergent series?

Answer: A series $\sum a_n$ is absolutely convergent if $\sum |a_n|$ converges. Stronger condition than simple convergence.

Flashcard 19: What does it mean for a series to be divergent?

Answer: The series does not approach a finite limit. Partial sums fail to approach any finite value.

Flashcard 20: Identify if the series $(-1)^n \frac{1}{\text{ln}(n+1)}$ converges.

Answer: Convergent. It satisfies the Alternating Series Test. Terms decrease in magnitude and approach zero.

Flashcard 21: Determine if $\sum \frac{1}{n(n+1)}$ is convergent.

Answer: Convergent. It is a telescoping series. Partial fraction telescopes to finite sum.

Flashcard 22: What is the $n$-th partial sum of the series $1 + \frac{1}{2} + \frac{1}{3} + \text{...}$?

Answer: $S_n = 1 + \frac{1}{2} + \frac{1}{3} + \text{...} + \frac{1}{n}$. Sum of first $n$ terms of the harmonic series.

Flashcard 23: Identify if the series $\frac{1}{n^{1.5}}$ converges.

Answer: Convergent. It is a p-series with $p = 1.5 > 1$. P-series converges when exponent exceeds 1.

Flashcard 24: What is the harmonic series' convergence status?

Answer: Divergent. Proven fact about this important series.

Flashcard 25: Identify the series: $1 + x + x^2 + x^3 + \text{...}$

Answer: Geometric series with first term $1$ and ratio $x$. Standard form with first term $a=1$ and ratio $r=x$.

Flashcard 26: State the condition for convergence using the Limit Comparison Test.

Answer: If $\lim_{n \to \infty} \frac{a_n}{b_n} = c > 0$, both series converge or diverge. Finite positive limit links convergence behavior.

Flashcard 27: Define a divergent series.

Answer: A series that does not approach a finite limit. Partial sums either grow without bound or oscillate.

Flashcard 28: What does it mean for a series to be divergent?

Answer: The series does not approach a finite limit. Partial sums fail to approach any finite value.

Flashcard 29: What is the P-Series Test?

Answer: A test used on series of form $\frac{1}{n^p}$ to determine convergence. Determines convergence for power series forms.

Flashcard 30: Determine if the series $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \text{...}$ converges.

Answer: Convergent. It is a geometric series with $r = \frac{1}{2}$. Since $|\frac{1}{2}| < 1$, the series converges.

Flashcard 31: Define the term 'partial sum' in the context of series.

Answer: The sum of the first $n$ terms of a series. Building block for determining series convergence.

Flashcard 32: What test would you use for the series $\frac{1}{n^2+n}$?

Answer: Comparison Test or Limit Comparison Test. Best suited for rational function series.

Flashcard 33: What is the Alternating Series Test?

Answer: A test to determine convergence of series with alternating signs. Applies to series of form $\sum (-1)^n a_n$.

Flashcard 34: State the Alternating Series Test conditions.

Answer: The terms decrease in absolute value and approach zero. Both conditions must hold for convergence.

Flashcard 35: State the integral test condition for convergence.

Answer: If $∫ f(x)dx$ converges, then $∑ a_n$ converges. Same convergence behavior as corresponding integral.

Flashcard 36: Define the term 'partial sum' in the context of series.

Answer: The sum of the first $n$ terms of a series. Building block for determining series convergence.

Flashcard 37: What is the integral test for series convergence?

Answer: A test using integrals to determine series convergence. Relates series convergence to improper integrals.

Flashcard 38: State the integral test condition for convergence.

Answer: If \text{∫} f(x)dx converges, then \text{∑} a_n converges. Same convergence behavior as corresponding integral.

Flashcard 39: State the condition for convergence using the Limit Comparison Test.

Answer: If $\lim_{n \to \infty} \frac{a_n}{b_n} = c > 0$, both series converge or diverge. Finite positive limit links convergence behavior.

Flashcard 40: Define a convergent series.

Answer: A series that approaches a finite limit as more terms are added. Partial sums approach a specific value as $n \to \infty$.

Flashcard 41: Determine if $\sum \frac{1}{n(n+1)}$ is convergent.

Answer: Convergent. It is a telescoping series. Partial fraction telescopes to finite sum.

Flashcard 42: What is the sum of a convergent geometric series $a + ar + ar^2 + \text{...}$?

Answer: $S = \frac{a}{1-r}$, where $|r| < 1$. Formula derived from infinite geometric series convergence.

Flashcard 43: What is the Root Test for convergence?

Answer: A test using $\lim_{n \to \infty} \sqrt[n]{|a_n|}$. Uses $n$-th root of terms for convergence analysis.

Flashcard 44: What is an absolutely convergent series?

Answer: A series $\sum a_n$ is absolutely convergent if $\sum |a_n|$ converges. Stronger condition than simple convergence.

Flashcard 45: What is the Ratio Test for series convergence?

Answer: A test using $\frac{a_{n+1}}{a_n}$ to determine convergence. Uses consecutive term ratios to test convergence.

Flashcard 46: Define a divergent series.

Answer: A series that does not approach a finite limit. Partial sums either grow without bound or oscillate.

Flashcard 47: When does the Root Test conclude convergence?

Answer: If $\lim_{n \to \infty} \sqrt[n]{|a_n|} < 1$. Limit less than 1 indicates series convergence.

Flashcard 48: For what values of $p$ does the series $\frac{1}{n^p}$ converge?

Answer: Converges if $p > 1$. Critical threshold separating convergence from divergence.

Flashcard 49: When is a series conditionally convergent?

Answer: When it converges, but its absolute value does not. Converges but not absolutely convergent.

Flashcard 50: Determine if the series $1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \text{...}$ is convergent or divergent.

Answer: Divergent. It is known as the harmonic series. Classic example of divergence despite decreasing terms.

Flashcard 51: When does the Ratio Test conclude convergence?

Answer: If $\lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| < 1$. Limit less than 1 guarantees series convergence.

Flashcard 52: What is the harmonic series?

Answer: The series $1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \text{...}$. Famous divergent series despite terms approaching zero.

Flashcard 53: What is the sum of a convergent geometric series $a + ar + ar^2 + \text{...}$?

Answer: $S = \frac{a}{1-r}$, where $|r| < 1$. Formula derived from infinite geometric series convergence.

Flashcard 54: What is an infinite series?

Answer: A sum of infinitely many terms, expressed as $a_1 + a_2 + a_3 + \text{...}$. The notation shows an endless sum of terms.

Flashcard 55: What is the harmonic series' convergence status?

Answer: Divergent. Proven fact about this important series.

Flashcard 56: Identify whether the series $1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \text{...}$ is convergent.

Answer: Convergent. It is an alternating series. Terms decrease and approach zero, satisfying AST.

Flashcard 57: Identify if the series $\sum \frac{1}{n!}$ converges.

Answer: Convergent. It is the exponential series. Factorial growth ensures rapid convergence.

Flashcard 58: Differentiate between conditional and absolute convergence.

Answer: Conditional convergence: series converges, but absolute value does not. Absolute convergence implies regular convergence.

Flashcard 59: What is the limit of the partial sums for a convergent series?

Answer: The finite value the series approaches. This limit exists and is finite for convergent series.

Flashcard 60: Identify if the series $\sum \frac{1}{n!}$ converges.

Answer: Convergent. It is the exponential series. Factorial growth ensures rapid convergence.

Flashcard 61: What is the sum of the series $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \text{...}$?

Answer: $1$. It is a geometric series with $a = \frac{1}{2}$, $r = \frac{1}{2}$. Using $S = \frac{a}{1-r} = \frac{1/2}{1-1/2} = 1$.