AP Calculus BC Flashcards: Comparison Tests For Convergence

What this deck covers

This deck focuses on Comparison Tests For Convergence, 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: Use Comparison: is $\frac{1}{n^2+2n}$ convergent?

Answer: Converges; compare to $\frac{1}{n^2}$ (convergent $p$-series). For large $n$, $\frac{1}{n^2+2n} \sim \frac{1}{n^2}$.

Flashcard 2: Does $\frac{1}{n}$ converge? Use the Comparison Test.

Answer: Diverges; compare to $\frac{1}{n}$ (divergent $p$-series). $\frac{1}{n}$ is a divergent $p$-series with $p = 1$.

Flashcard 3: Identify a series to compare with $\frac{1}{n^4+3n^2}$.

Answer: Compare with $\frac{1}{n^4}$, a convergent $p$-series. For large $n$, $\frac{1}{n^4+3n^2} \sim \frac{1}{n^4}$.

Flashcard 4: State the Limit Comparison Test.

Answer: If $\frac{a_n}{b_n} \to c > 0$, both series converge or diverge. Both series share the same convergence behavior when the limit exists and is positive.

Flashcard 5: State the condition for the Limit Comparison Test to be valid.

Answer: If $\frac{a_n}{b_n} \to c > 0$, it is valid. This ensures both series have the same convergence behavior.

Flashcard 6: In the Comparison Test, what do you compare $a_n$ to?

Answer: Compare $a_n$ to $b_n$ where $b_n$ is a known series. The known series $b_n$ serves as the reference for comparison.

Flashcard 7: For Limit Comparison, what happens if $c = 0$ or $c = \text{infinity}$?

Answer: The test is inconclusive if $c = 0$ or $c = \text{infinity}$. These limit values don't provide definitive comparison information.

Flashcard 8: What is required for the Comparison Test to be applicable?

Answer: Series terms $a_n$ and $b_n$ must be positive for all $n$. Negative terms invalidate the comparison inequalities used in the test.

Flashcard 9: Use Comparison: is $\frac{n^2}{n^3+1}$ convergent?

Answer: Diverges; compare to $\frac{1}{n}$ (divergent $p$-series). For large $n$, $\frac{n^2}{n^3+1} \sim \frac{1}{n}$.

Flashcard 10: When using the Comparison Test, what must be true of $b_n$?

Answer: $b_n$ must be a series with known convergence behavior. Without known behavior, no conclusion can be drawn.

Flashcard 11: Determine convergence of $\frac{1}{n^{2.5}}$ using Comparison.

Answer: Converges; compare to $\frac{1}{n^{2.5}}$ (convergent $p$-series). $p = 2.5 > 1$ makes this a convergent $p$-series.

Flashcard 12: Use Comparison: is $\frac{1}{\text{ln}(n) \times n}$ convergent?

Answer: Diverges; compare to $\frac{1}{n}$ (divergent $p$-series). Since $\ln(n)$ grows slower than any power, this behaves like $\frac{1}{n}$.

Flashcard 13: For Limit Comparison, what if $a_n$ and $b_n$ are non-positive?

Answer: Test is not applicable; terms must be positive. The test relies on positive term inequalities.

Flashcard 14: Use Limit Comparison: compare $\frac{n}{n^2+1}$ with $\frac{1}{n}$.

Answer: Diverges; limit is finite and positive. $\lim_{n \to \infty} \frac{n}{n^2+1} \cdot \frac{n}{1} = 1 > 0$.

Flashcard 15: Use Comparison: does $\frac{n^2 + 3}{n^3 + 1}$ converge?

Answer: Diverges; compare to $\frac{1}{n}$ (divergent $p$-series). For large $n$, $\frac{n^2+3}{n^3+1} \sim \frac{1}{n}$.

Flashcard 16: Use Limit Comparison: compare $\frac{1}{n^2+1}$ with $\frac{1}{n^2}$.

Answer: Converges; limit is finite and positive. $\lim_{n \to \infty} \frac{1}{n^2+1} \cdot \frac{n^2}{1} = 1 > 0$.

Flashcard 17: Use Limit Comparison: compare $\frac{n^2+1}{n^4}$ with $\frac{1}{n^2}$.

Answer: Converges; limit is finite and positive. $\lim_{n \to \infty} \frac{n^2+1}{n^4} \cdot \frac{n^2}{1} = 1 > 0$.

Flashcard 18: Identify a series to compare with $\frac{3}{n^3 + \text{ln}(n)}$.

Answer: Compare with $\frac{1}{n^3}$, a convergent $p$-series. For large $n$, $\frac{3}{n^3 + \ln(n)} \sim \frac{3}{n^3}$.

Flashcard 19: Does the Limit Comparison Test require non-negative terms?

Answer: Yes, terms $a_n$ and $b_n$ must be positive. Positivity is essential for meaningful ratio comparison.

Flashcard 20: What $p$-series can you use to compare when $p > 1$?

Answer: Use $\frac{1}{n^p}$, which converges for $p > 1$. $p$-series with $p > 1$ are standard convergent comparison series.

Flashcard 21: Can the Limit Comparison Test be used if $c < 0$?

Answer: No, $c$ must be positive for the Limit Comparison Test. Negative limits don't maintain the required proportional relationship.

Flashcard 22: What is the relationship between $a_n$ and $b_n$ in Limit Comparison?

Answer: If $\frac{a_n}{b_n} \to c > 0$, both series behave similarly. When $c > 0$, the series have proportional terms and same convergence.

Flashcard 23: Can the Comparison Test be used for alternating series?

Answer: No, the test requires positive terms. Alternating series have both positive and negative terms.

Flashcard 24: Is $\frac{1}{n^{1.5}}$ convergent? Use Comparison.

Answer: Converges; compare to $\frac{1}{n^{1.5}}$ (convergent $p$-series). $p = 1.5 > 1$ makes this a convergent $p$-series.

Flashcard 25: Identify if $\frac{1}{n^3 + n}$ is convergent using Comparison.

Answer: Converges; compare to $\frac{1}{n^3}$ (convergent $p$-series). For large $n$, $\frac{1}{n^3 + n} \sim \frac{1}{n^3}$.

Flashcard 26: Identify a series to compare with $\frac{2n^2+1}{n^3+3}$.

Answer: Compare with $\frac{1}{n}$, a divergent $p$-series. For large $n$, $\frac{2n^2+1}{n^3+3} \sim \frac{2}{n}$.

Flashcard 27: Identify if $\frac{1}{n^2}$ converges using the Comparison Test.

Answer: Converges; compare to $\frac{1}{n^2}$ (convergent $p$-series). $\frac{1}{n^2}$ is itself a convergent $p$-series with $p = 2 > 1$.

Flashcard 28: What is the Comparison Test for convergence?

Answer: A test comparing a series to a known convergent or divergent series. Direct comparison determines convergence by relating to known series.

Flashcard 29: Use Comparison: does $\frac{1}{n^{1.1}}$ converge?

Answer: Converges; compare to $\frac{1}{n^{1.1}}$ (convergent $p$-series). $p = 1.1 > 1$ makes this a convergent $p$-series.

Flashcard 30: Which series should you choose for the Comparison Test?

Answer: Choose a series $b_n$ that is similar in form to $a_n$. Similar form makes the comparison meaningful and easier to evaluate.