Power Series: Radius, Interval of Convergence - AP Calculus BC

1. Determine the interval of convergence for $\textstyle\sum_{n=1}^{\infty} \frac{(-1)^n x^n}{n}$.

   $(-1, 1]$. Alternating series converges at $x = 1$; diverges at $x = -1$.

2. What is the interval of convergence for a power series?

   Values of $x$ where the series converges. The set of all $x$-values where the power series converges.

3. What does the radius of convergence $R = \text{∞}$ imply?

   Series converges for all $x$. Infinite radius means convergence for every real number.

4. State the formula for the root test applied to a series.

   $\textstyle\lim_{n \to \infty} \big|a_n\big|^{1/n} < 1$ for convergence. Root test condition for series convergence.

5. For which $x$ does the series $\textstyle\sum_{n=0}^{\infty} a_n x^n$ converge absolutely?

   $|x| < R$. Within radius of convergence, series converges absolutely.

6. Identify the radius of convergence for $\sum_{n=0}^{\infty} 2^n x^n$.

   $R = \frac{1}{2}$. Rewrite as $\sum (2x)^n$; radius is $\frac{1}{2}$.

7. What does it mean if a power series diverges for $x = c + R$?

   Endpoint $x = c + R$ not in interval of convergence. Endpoint divergence excludes that point from the convergence interval.

8. Identify the radius of convergence for $a_n = \frac{1}{n^2}$ using the ratio test.

   $R = \text{∞}$. Quadratic decay gives infinite radius via ratio test.

9. Identify the radius of convergence if $R = 0$ for a series.

   Series converges only at $x = c$. Zero radius means convergence only at the center point.

10. Identify the radius of convergence for a geometric series $\sum x^n$.

    $R = 1$. Standard geometric series has radius 1.

11. Find the radius of convergence for $\textstyle\bigsum_{n=1}^{\text{∞}} x^n$ using the root test.

    $R = 1$. Root test gives $\lim |x| = |x|$; radius is 1.

12. What is the radius of convergence if $L = 0$ in the ratio test?

    $R = \text{∞}$. When limit is 0, radius becomes infinite.

13. Identify the radius of convergence for $\textstyle \sum_{n=0}^{\infty} 5^n x^n$.

    $R = \frac{1}{5}$. Rewrite as $\sum (5x)^n$; radius is $\frac{1}{5}$.

14. Which condition indicates absolute convergence of a series?

    If $\textstyle\bigsum |a_n|$ converges. Absolute convergence means the series of absolute values converges.

15. Find the radius of convergence for $\textstyle\bigsum_{n=1}^{\text{∞}} \frac{x^n}{n^3}$.

    $R = \text{∞}$. Cubic decay gives infinite radius via ratio test.

16. Find the interval of convergence for $\textstyle\bigsum_{n=0}^{\text{∞}} \frac{x^n}{n!}$.

    $(-\text{∞}, \text{∞})$. Exponential function series; factorial dominates, giving infinite radius.

17. Determine the radius of convergence for $\sum_{n=0}^{\infty} \frac{(2x)^n}{n!}$.

    $R = \infty$. Factor out constants; factorial dominates for infinite radius.

18. Identify the convergence interval for $\textstyle\bigsum_{n=0}^{\text{∞}} x^n$ using the ratio test.

    $(-1, 1)$. Standard geometric series; endpoints both diverge.

19. What test can be used to determine the convergence of a power series?

    Ratio test or root test. Standard convergence tests for determining series behavior.

20. What is the interval of convergence for $\textstyle \sum_{n=1}^{\infty} \frac{x^n}{n^2}$?

    $(-\infty, \infty)$. $p$-series with $p = 2 > 1$ converges; infinite radius for power series.

21. For which $x$ does the series $\sum_{n=0}^{\infty} a_n x^n$ converge absolutely?

    $|x| < R$. Within radius of convergence, series converges absolutely.

22. Find the radius of convergence for $a_n = n^2$ using the root test.

    $R = 0$, series converges only at $x = c$. Quadratic growth makes $\lim |a_n|^{1/n} = \infty$, so $R = 0$.

23. Determine the convergence interval of $\textstyle\sum_{n=1}^{\infty} \frac{x^n}{n}$.

    $(-1, 1]$. Alternating harmonic-type series; converges at $x = 1$ by alternating series test.

24. Determine the interval of convergence for $\textstyle\bigsum_{n=0}^{\text{∞}} \frac{(x-3)^n}{2^n}$.

    $(1, 5)$. Series $\sum \frac{(x-3)^n}{2^n}$ has center 3, radius 2; endpoints diverge.

25. Find the radius of convergence of $\textstyle\sum_{n=1}^{\infty} \frac{x^n}{3^n}$.

    $R = 3$. Rewrite as $\sum (\frac{x}{3})^n$; geometric series with ratio $\frac{1}{3}$.

26. Determine the convergence interval of $\textstyle\bigsum_{n=1}^{\infty} \frac{x^n}{n}$

    $(-1, 1]$. Alternating harmonic-type series; converges at $x = 1$ by alternating series test.

27. State the interval of convergence for a series with $R = 0$.

    $\text{x = c}$ only. Zero radius restricts convergence to the center point only.

28. Which test confirms divergence if the limit is greater than 1?

    Ratio test or root test. Both tests indicate divergence when limit exceeds 1.

29. Identify the radius of convergence for $\textstyle\bigsum_{n=0}^{\text{∞}} 5^n x^n$.

    $R = \frac{1}{5}$. Rewrite as $\sum (5x)^n$; radius is $\frac{1}{5}$.

30. State the formula for determining convergence using the root test.

    $\textstyle \lim_{n \to \infty} \big| a_n \big|^{1/n} < 1$. Root test convergence criterion.

31. What is the radius of convergence when $L = 1$ in the ratio test?

    Test is inconclusive. When $L = 1$, ratio test gives no information about convergence.

32. What is the interval of convergence for $\textstyle \sum_{n=0}^{\infty} \frac{x^n}{n!}$?

    $(- \infty, \infty)$. Exponential function has infinite radius of convergence.

33. Determine the interval of convergence for the series $\textstyle \sum_{n=0}^{\infty} x^n$.

    $-1, 1$. Geometric series with ratio $|x| < 1$; endpoints diverge.

34. What does $L = 1$ imply in the ratio test for convergence?

    Test is inconclusive. Ratio test fails when limit equals 1; other methods needed.

35. What is the interval of convergence for $\textstyle\bigsum_{n=1}^{\text{∞}} \frac{x^n}{n^2}$?

    $(-\text{∞}, \text{∞})$. $p$-series with $p = 2 > 1$ converges; infinite radius for power series.

36. Determine the interval of convergence for $\textstyle\sum_{n=1}^{\infty} \frac{(-1)^n x^n}{n}$

    $(-1, 1]$. Alternating series converges at $x = 1$; diverges at $x = -1$.

37. What does $L < 1$ indicate in the ratio test for a series?

    Series converges absolutely. Ratio test shows absolute convergence when limit is less than 1.

38. What is the general form of a power series centered at $c$?

    $\text{Series: } \textstyle\sum_{n=0}^{\infty} a_n(x - c)^n$. Standard power series form with center $c$ and coefficients $a_n$.

39. Determine the interval of convergence for $\sum_{n=1}^{\infty} \frac{x^n}{n}$ using the ratio test.

    $(-1, 1]$. Series $\sum \frac{x^n}{n}$ converges at $x = 1$ by alternating series test.

40. Identify the radius of convergence for $\textstyle\bigsum_{n=0}^{\text{∞}} 2^n x^n$.

    $R = \frac{1}{2}$. Rewrite as $\sum (2x)^n$; radius is $\frac{1}{2}$.

41. Identify the radius of convergence for a geometric series $\textstyle\bigsum x^n$.

    $R = 1$. Standard geometric series has radius 1.

42. What is the interval of convergence for $\textstyle\sum_{n=0}^{\infty} \frac{x^n}{n!}$?

    $(-\infty, \infty)$. Exponential function has infinite radius of convergence.

43. What is the radius of convergence if $L = \text{∞}$ in the ratio test?

    $R = 0$. Infinite limit in ratio test means zero radius.

44. What condition implies divergence in the root test?

    $\textstyle \lim_{n \to \infty} \big| a_n \big|^{1/n} > 1$. Root test divergence condition.

45. Find the radius of convergence for $a_n = \frac{1}{n!}$ using the ratio test.

    $R = \text{∞}$, series converges for all $x$. Factorial growth dominates, making the limit 0, so $R = \frac{1}{0} = \infty$.

46. State the formula for the radius of convergence using the ratio test.

    $R = \frac{1}{L}$ where $L = \textstyle\lim_{n \to \infty} \bigg| \frac{a_{n+1}}{a_n} \bigg|$. Ratio test formula where $L$ is the limit of consecutive coefficient ratios.

47. Determine the interval of convergence for the series $\sum_{n=0}^{\infty} x^n$.

    $(-1, 1)$. Geometric series with ratio $|x| < 1$; endpoints diverge.

48. Determine the interval of convergence for $\textstyle\sum_{n=1}^{\infty} \frac{x^n}{n}$ using the ratio test.

    $(-1, 1]$. Series $\sum \frac{x^n}{n}$ converges at $x = 1$ by alternating series test.

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