Defining Convergent and Divergent Infinite Series - AP Calculus BC

1. What is the Limit Comparison Test?

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

2. What is a telescoping series?

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

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

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

4. When does the Ratio Test conclude convergence?

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

5. State the condition for a geometric series to be convergent.

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

6. What does it mean for a series to be divergent?

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

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

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

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

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

9. Identify if the series $\frac{1}{n^{1.5}}$ converges.

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

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

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

11. Define a divergent series.

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

12. What is the P-Series Test?

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

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

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

14. What test would you use for the series $\frac{1}{n^2+n}$?

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

15. What is the Alternating Series Test?

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

16. State the Alternating Series Test conditions.

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

17. Define the term 'partial sum' in the context of series.

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

18. What is the integral test for series convergence?

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

19. State the integral test condition for convergence.

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

20. State the condition for convergence using the Limit Comparison Test.

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

21. Define a convergent series.

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

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

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

23. What is the Root Test for convergence?

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

24. What is an absolutely convergent series?

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

25. What is the Ratio Test for series convergence?

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

26. When does the Root Test conclude convergence?

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

27. For what values of $p$ does the series $\frac{1}{n^p}$ converge?

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

28. When is a series conditionally convergent?

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

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

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

30. What is the harmonic series?

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

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

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

32. What is an infinite series?

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

33. What is the harmonic series' convergence status?

    Divergent. Proven fact about this important series.

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

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

35. Differentiate between conditional and absolute convergence.

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

36. What is the limit of the partial sums for a convergent series?

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

37. Identify if the series $\sum \frac{1}{n!}$ converges.

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

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

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

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