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 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 2: What is the integral test for series convergence?

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

Flashcard 3: 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 4: 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 5: State the Alternating Series Test conditions.

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

Flashcard 6: 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 7: 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 $S = \frac{a}{1-r} = \frac{1/2}{1-1/2} = 1$.

Flashcard 8: 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 9: 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 10: 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 11: 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 12: 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 13: Identify if the series $\sum \frac{1}{n!}$ converges.

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

Flashcard 14: 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 15: 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 16: 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 17: 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 18: 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 19: What is the harmonic series' convergence status?

Answer: Divergent. Proven fact about this important series.

Flashcard 20: 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 21: Differentiate between conditional and absolute convergence.

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

Flashcard 22: What is a telescoping series?

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

Flashcard 23: 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 24: 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 25: 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 26: 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 27: 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 28: 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 29: Define a divergent series.

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

Flashcard 30: Define a convergent series.

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