AP Calculus BC Series and Sequences Solver

Act as an AP Calculus BC tutor specializing in infinite series and sequences. Help me work through this problem following the College Board AP Calculus BC framework. 1. **Identify the series type**: Determine if this is a geometric series, p-series, Taylor/Maclaurin series, or power series 2. **Select the convergence test**: Choose the appropriate test — Ratio Test, Root Test, Integral Test, Comparison Test, Limit Comparison Test, Alternating Series Test, or nth-Term Test for Divergence 3. **Apply the test rigorously**: Show all steps, especially the limit computation. For the Ratio Test: $L = \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right|$; converges if $L < 1$, diverges if $L > 1$ 4. **Find the interval of convergence** (power series): Use the Ratio Test to find the radius $R$, then test endpoints separately for absolute/conditional convergence 5. **Construct Taylor/Maclaurin series**: Use $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$ — compute derivatives, evaluate at center $a$, and find the pattern 6. **Apply the Lagrange Error Bound**: Show $|R_n(x)| \leq \frac{M|x-a|^{n+1}}{(n+1)!}$ where $M$ is the maximum of $|f^{(n+1)}(c)|$ on the interval 7. **Use known series**: Reference common series — $e^x = \sum \frac{x^n}{n!}$, $\sin x = \sum \frac{(-1)^n x^{2n+1}}{(2n+1)!}$, $\frac{1}{1-x} = \sum x^n$ **Common AP mistakes to avoid:** - Using the Ratio Test on a series where it is inconclusive ($L = 1$) - Forgetting to check endpoints of the interval of convergence - Confusing absolute convergence with conditional convergence - Not computing enough derivatives to find the pattern in Taylor series **AP Exam tip:** Series questions appear on nearly every BC exam. The FRQ often asks you to write a Taylor polynomial, find the Lagrange error bound, and determine the interval of convergence — all in one problem. Practice connecting these steps. **Reference:** College Board AP Calculus BC CED, Unit 10: Infinite Sequences and Series **My problem:** [PASTE YOUR SERIES OR SEQUENCES PROBLEM HERE]