AP Calc BC Part 2 Notes

Here are the LaTeX math equations formatted correctly with the appropriate syntax:

1. Arithmetic Sequence Term: $t_n = a + (n-1)d$ (where $a$ is the first term and $d$ is the common difference)

2. Sum of an Arithmetic Series: $S_n = \frac{n}{2}(2a + (n-1)d)$

3. Geometric Sequence Term: $t_n = a r^{n-1}$ (where $a$ is the first term and $r$ is the common ratio)

4. Sum of a Finite Geometric Series: $S_n = a\frac{1 - r^n}{1 - r}$

5. Sum of an Infinite Geometric Series: $S = \frac{a}{1-r}$, provided that |r| < 1

6. Absolute Convergence: Both the original series $\sum a_n$ and the absolute value series $\sum |a_n|$ converge.

7. Conditional Convergence: The original series $\sum a_n$ converges, but the absolute value series $\sum |a_n|$ diverges.

8. Integral Test Conditions: If $f(x)$ is positive, continuous, and decreasing for $x \\ge 1$ where $f(k) = a_k$, then $\int_{1}^{\infty} f(x) \, dx$ and $\sum_{k=1}^{\infty} a_k$ either both converge or both diverge.

9. P-Series Test: $\sum_{k=1}^{\infty} \frac{1}{k^p}$ converges if p > 1; diverges if 0 < p \le 1.

10. Direct Comparison Test: If $0 \le a_n \le b_n$, then if $\sum b_k$ converges, so does $\sum a_k$. If $\sum a_k$ diverges, then $\sum b_k$ diverges as well.

11. Ratio Test: Evaluate $L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right|$. If L < 1, the series converges absolutely; if L > 1, it diverges; if $L = 1$, the test is inconclusive.

12. n-th Term Test for Divergence: If $\lim_{k \to \infty} a_k \neq 0$, then the series $\sum a_k$ diverges.

13. Alternating Series Test Conditions: 1. a_k > 0 for all $k$.

14. $\lim_{k \to \infty} a_k = 0$.

15. $a_{k+1} \le a_k$ (the terms are non-increasing).

16. Geometric Series Test: $\sum a r^n$ converges if |r| < 1; diverges if $|r| \ge 1$.

17. Limit Comparison Test: Evaluate $c = \lim_{k \to \infty} \frac{a_k}{b_k}$. If 0 < c < \infty, then both series either both converge or diverge.

18. General Maclaurin Series: $\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n$

19. Maclaurin Series for sin(x): $\sin(x) = \sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)!} x^{2k+1}$

20. Maclaurin Series for cos(x): $\cos(x) = \sum_{k=0}^{\infty} \frac{(-1)^k}{(2k)!} x^{2k}$

21. Maclaurin Series for e^x: $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$

22. Maclaurin Series for \frac{1}{1-x}: $\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n$ valid for |x| < 1.

23. General Taylor Series: $\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n$

24. Alternating Series Error Bound: $|S - S_k| \le a_{k+1}$.

25. Lagrange Error Bound: $|E_n(x)| \le \frac{M}{(n+1)!} |x-a|^{n+1}$.

26. Interval of Convergence Three Possibilities:

    1. Converges only at $x = a$ ($R = 0$)

    2. Converges absolutely for all $x$ ($R = \infty$)

    3. Converges for |x-a| < R and diverges for |x-a| > R, yielding interval $(a-R, a+R)$.

27. Steps to Find Interval of Convergence:

    1. Apply the Ratio Test to solve for |x-a| < R.

    2. Test each endpoint $x = a-R$ and $x = a+R$ for convergence.

28. First Derivative for Parametric Equations: $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$.

29. Second Derivative for Parametric Equations: $\frac{d^2y}{dx^2} = \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}$.

30. Rectangular Coordinate Equations for Polar Coordinates: $x = r \cos(\theta)$, $y = r \sin(\theta)$, and $x^2 + y^2 = r^2$.

31. Polar Curve Derivative: $\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} = \frac{\frac{dr}{d\theta}\sin(\theta) + r\cos(\theta)}{\frac{dr}{d\theta}\cos(\theta) - r\sin(\theta)}$.

32. Area Bounded by a Polar Curve: $A = \frac{1}{2} \int_{a}^{b} r^2 \, d\theta$.

33. Area Between Two Polar Curves: $A = \frac{1}{2} \int_{a}^{b} \left((r_2)^2 - (r_1)^2\right) d\theta$.

34. Cartesian Arc Length: $L = \int_{a}^{b} \sqrt{1 + [f'(x)]^2} \, dx$ (for $y = f(x)$).

35. Cartesian Arc Length for $x = g(y)$: $L = \int_{c}^{d} \sqrt{1 + [g'(y)]^2} \, dy$.

36. Parametric Arc Length: $L = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt$.