Comparison Tests for Convergence - AP Calculus BC

1. Does the Limit Comparison Test require non-negative terms?

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

2. State the Limit Comparison Test.

   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.

3. State the condition for the Limit Comparison Test to be valid.

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

4. In the Comparison Test, what do you compare $a_n$ to?

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

5. What $p$-series can you use to compare when $p > 1$?

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

6. Use Comparison: is $\frac{1}{n^{0.5}}$ convergent?

   Diverges; compare to $\frac{1}{n^{0.5}}$ (divergent $p$-series). $p = 0.5 < 1$ makes this a divergent $p$-series.

7. When using the Comparison Test, what must be true of $b_n$?

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

8. What is the Comparison Test for convergence?

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

9. Which series should you choose for the Comparison Test?

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

10. What is required for the Comparison Test to be applicable?

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

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

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

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

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

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

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

14. Find if $\frac{1}{n^3}$ converges using the Comparison Test.

    Converges; compare to $\frac{1}{n^2}$ (convergent $p$-series). $\frac{1}{n^3}$ is a convergent $p$-series since $p = 3 > 1$.

15. For Limit Comparison, what happens if $c = 0$ or $c = \text{infinity}$?

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

16. Determine convergence of $\frac{1}{n^{2.5}}$ using Comparison.

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

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

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

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

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

19. Can the Comparison Test be used for alternating series?

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

20. Does $\frac{1}{n}$ converge? Use the Comparison Test.

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

21. Use Comparison: does $\frac{1}{n^{1.1}}$ converge?

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

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

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

23. Can the Limit Comparison Test be used if $c < 0$?

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

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

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

25. For Limit Comparison, what if $a_n$ and $b_n$ are non-positive?

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

26. Find if $\frac{1}{n^2 + \text{ln}(n)}$ converges using Comparison.

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

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

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

28. Use Comparison: is $\frac{1}{n^2+2n}$ convergent?

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

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

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

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

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

31. Is $\frac{1}{n^{1.5}}$ convergent? Use Comparison.

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

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

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

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

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

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