Representing Series as Power Series - AP Calculus BC

1. Determine the radius of convergence for $\text{Sum} = (2x)^n$.

   $R = \frac{1}{2}$. Series $\sum (2x)^n$ has radius $\frac{1}{2}$ from ratio test.

2. What is the interval of convergence for $1 + x + x^2 + \text{...}$?

   $|x| \text{ \textless 1}$. Basic geometric series convergence condition.

3. Convert $\text{Sum} = \frac{x^n}{3^n}$ into a power series centered at 1.

   $\text{Sum} = \frac{(x-1)^n}{3^n}$. Substitute $(x-1)$ for $x$ in the original series.

4. What is the power series for $\frac{1}{1+x}$?

   $\text{Sum} = (-1)^n x^n$ for $|x| < 1$. Geometric series with alternating signs from ratio $-x$.

5. Identify the power series for $\frac{1}{1-2x}$.

   $\text{Sum} = (2x)^n$ for $|2x| < 1$. Geometric series with ratio $2x$ requires $|2x| < 1$.

6. What is the Maclaurin series for $\text{sin}(x)$?

   $\text{Sum} = (-1)^n \frac{x^{2n+1}}{(2n+1)!}$. Maclaurin series for sine with odd powers and alternating signs.

7. Identify the power series for $\frac{1}{1-x^2}$.

   $\text{Sum} = x^{2n}$ for $|x| \text{ \textless 1}$. Geometric series with $x^2$ substitution for even powers only.

8. Determine the interval of convergence for $\text{Sum} = \frac{x^n}{n^3}$.

   $|x| < 1$. P-series with $p=3>1$ converges when $|x| < 1$.

9. Find the interval of convergence for $\text{Sum} = \frac{x^n}{n}$.

   $|x| < 1$. Harmonic-like series converges when $|x| < 1$.

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

    $\sum a_n (x-c)^n$. Standard form where coefficients multiply powers of $x - c$.

11. State the power series for $\tan^{-1}(x)$.

    $\text{Sum} = (-1)^n \frac{x^{2n+1}}{2n+1}$ for $|x| \text{ \textless 1}$. Alternating series with odd powers and odd factorial-like denominators.

12. Convert the series $\text{Sum} = \frac{x^n}{4^n}$ to a power series centered at 2.

    $\text{Sum} = \frac{(x-2)^n}{4^n}$. Substitute $(x-2)$ for $x$ in series $\sum \frac{x^n}{4^n}$.

13. What is the interval of convergence for $\text{Sum} = \frac{x^n}{n^2}$?

    $|x| \text{ \textless 1}$. P-series with $p=2>1$ converges when $|x| < 1$.

14. State the formula for the interval of convergence of a power series.

    $|x-c| < R$. Distance from center must be less than radius of convergence.

15. State the power series for $\text{ln}(1+x)$.

    $\text{Sum} = (-1)^{n+1} \frac{x^n}{n}$ for $|x| \text{ \textless 1}$. Alternating harmonic series with convergence $|x| < 1$.

16. Find the radius of convergence for $\text{Sum} = \frac{x^n}{n!}$.

    $R = \text{infinity}$. Exponential series converges for all real values.

17. Determine the power series for $e^{-x}$.

    $\text{Sum} = (-1)^n \frac{x^n}{n!}$. Exponential series with $-x$ substituted for $x$.

18. Find the power series representation for $\frac{1}{(1-x)^2}$.

    $\text{Sum} = (n+1)x^n$ for $|x| < 1$. Derivative of geometric series gives powers with coefficients.

19. What is the Taylor series for $e^x$ centered at $c=0$?

    $\text{Sum} = \frac{x^n}{n!}$. Maclaurin series for exponential function at origin.

20. Convert the series $\text{Sum} = x^n$ to a power series centered at 2.

    $\text{Sum} = (x-2)^n$. Substitute $(x-2)$ for $x$ in original series.

21. What is the radius of convergence for $\sum x^n$?

    $R = 1$. Geometric series $\sum x^n$ has unit radius.

22. Identify the power series for $\frac{1}{1+x^2}$.

    $\text{Sum} = (-1)^n x^{2n}$ for $|x| \text{ \textless 1}$. Geometric series with $x^2$ and alternating signs.

23. What is the radius of convergence for $\text{Sum} = x^n$?

    $R = 1$. Geometric series $\sum x^n$ has unit radius.

24. What is the Taylor series for $\text{cos}(x)$ centered at $c=0$?

    $\text{Sum} = (-1)^n \frac{x^{2n}}{(2n)!}$. Maclaurin series for cosine with even powers and alternating signs.

25. Which test can determine the convergence of a power series?

    The Ratio Test. Uses ratios of consecutive terms to find convergence radius.

26. Identify the power series for $\frac{1}{1-x}$.

    $\text{Sum} = x^n$ for $|x| < 1$. Geometric series formula with first term 1 and ratio $x$.

27. Convert $\text{Sum} = x^n$ to a power series centered at $c=5$.

    $\text{Sum} = (x-5)^n$. Shift power series by substituting $(x-5)$ for $x$.

28. What is the power series representation for $\text{ln}(1-x)$?

    $\text{Sum} = -\frac{x^n}{n}$ for $|x| \text{ \textless 1}$. Natural log series with $-x$ substituted for $x$.

29. Convert the series $\text{Sum} = \frac{x^n}{n!}$ to a power series centered at 3.

    $\text{Sum} = \frac{(x-3)^n}{n!}$. Shift exponential series by substituting $(x-3)$ for $x$.

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